As the others say, nothing material that we know of. Just far hotter than temperatures that permits anything to be solid. But there are a few things to explore.
First refrigeration. Okay the sun is far too hot for any material we can construct. But what if we are able somehow to dissipate all that heat?
If we can somehow reflect the heat away, then our materials will never get hot. So - if you had a perfect reflector, doesn't matter how hot it gets.
But - sadly - we have dense, very hot materials in direct contact with our "heat mirror" - hard to see that working. If we could somehow create a perfect mirror - it could reflect away the heat radiation - but how could it reflect away the heat from the bombarding atoms of the surrounding material?
I can think of one way to do it - but it is both extremely dangerous for civilization, and also - hard to imagine it actually working. But suppose we could create a Mini black hole (if such things exist), of substantial size - then it could act as a heat sink because heat going into the black hole would not come back out again. Mini blackholes, if they exist, many physicists think, dissipate due to Hawking radiation. But if you could keep it above that size, just large enough to not produce a significant amount of radiation, then you'd be okay.
I don't recommend this though, what if the black hole escapes from our spaceship, then it is goodbye sun :).
Or for that matter, doesn't need to be a mini black hole. Make your heat sink out of any super dense material.
Main thing is to start up cold before you go in, and to have enough material in it to absorb all the heat for your transition through the sun.
It doesn't matter if it melts. That would be like the ablative layer on heat shields for re-entry. Indeed if it melts away - and then evaporates - that helps to absorb more heat.
This seems worth looking at closer, see later.
You could surround yourself with neutron star material. And maybe if it starts off cold, and you go through the sun quickly, you can get through to the other side without heating up significantly.
That is if we can find a way to keep neutron star material in a solid state without it exploding, during the transition. And form it into a shell that you can enter without being turned into spaghetti as you go into the shell.
A shell of super dense material would be okay even if externally it had even several Earth's masses of material and high g from the outside - if you are inside it, you will experience no gravitational force at all. That is also - supposing it is constructed to be perfectly spherical to an extraordinary level of precision. Then you would feel no gravity inside - if you can get inside it in the first place that is.
You could just make it with a hole in the side and you "fall into it" and then rapidly seal the hole. But still, you'd have the tidal forces. As you fall into your protective shelter, at certain times during your fall your feet would be accelerating far more rapidly than your head. And you'd turn into spaghetti. So I think, a bit difficult to "don" this spacesuit, if it is possible at all.
I think what you'd have to do is to build it around you instead. That is if this material is possible at all, more later.
Just for completeness. Don't have to absorb the heat if you can pump it away. I.e. use some other source of power to pump the heat from inside to out using a power source such as electricity. See Thermoelectric cooling - you can cool down spaceships to well below the ambient temperature in space from incoming radiation.
If you could get that to work, and also had an almost unlimited source of power, e.g. fusion power, you could stay in the sun indefinitely, as long as the power source lasts. But - though it works fine for spaceships, could this work in the sun?...
As for the gravity of the sun - if you can shield from the heat - and also - maintain your velocity, you can keep the interior of the spaceship at any desired gravity.
So e.g. if you had a powerful motor programmed to compensate for the resistance of the material you are going through so that it feels like free flight to the residents of the spaceship, you'd be in zero g even on a flight right through the center of the sun. And could come out the other side. For that matter you could also accelerate at 1 g all the way through and come out far faster than you went in, and be in comfort all the way through.
You could even - in a series of orbits, if you can keep the heat away, you could spiral down and end up at the center of the sun and you'd experience no g force there at all of course. You'd be floating in zero g.
Fusion reactors - they are close to being able to contain the material for fusion, for long enough for them to fuse, at temperatures far higher than the temperature of the sun. But the problem there is - that it is at very low densities. And also for short time periods.
Could future physics would permit them to hold back material at densities of the center of the sun?
As gravity is not an issue, if we had the "force field" of science fiction, it would be no problem.
Of all the ideas - I think the idea of a heat sink made of some more ordinary material at least is worth exploring. One that is dense enough to absorb all the heat of your transition.
How much heat would it need to absorb? How much mass is needed to absorb all that heat, supposing it starts off at around absolute zero?
Maybe worth doing a bit of a calculation here....
Note, that we can absorb more heat also if it is designed as an ablative layer so that the outside melts and then evaporates.
Problem there though is - that the outside is far hotter than humans could tolerate. But perhaps could have an external layer that ablates, to get rid of much of the heat in the initial phases - and then an inner layer insulated from it that just absorbs heat - and then the humans inside that.
That way we could get rid of much more heat than we can by just absorbing it in a heat sink.
But I'll ignore that for now, let's see what if we just have an ordinary heat sink, all at same temperature, with humans at centre.
Of course some of the exterior of it will melt and ablate - and that will add to its heat sink capabilities, but not designed as an ablative layer as such all the way through.
Just want some kind of rough back of the envelope calculation here, is it possible at all?
I'm expecting our "spacesuit" to be rather large, kilometers, maybe moon sized, maybe larger. But - it does seem to be at least feasible.
Example. Suppose that we used Mercury as our "spacesuit" with some future technology that permits us to first cool Mercury down to absolute zero, all the way to its core, and then fly Mercury right through the centre of the sun, with rocket motors so powerful we can maintain an artificial "zero g" flight throughout - with the humans inside it?
Surely that would work (without doing the calculation, but seems unlikely that a few hours passage through the sun would make it too hot for humans to tolerate). So - could we make do with something a bit smaller?
Or would it work?
So, let's see.
The sun's radius is 695842 km. So - traveling at zero g, you could get through it in half an orbit of a planet with semi-major diameter of 695842 km.
Using Orbital period of a planet (online calculator) I make the orbital period 2.78043 hours. So you can get through the sun in zero g in about an hour and a half easily.
You'd get through it somewhat more quickly if accelerating at full g all the way. But I don't think that will make a significant difference here since most of the time we are traveling at the full g of the sun.
On the other hand in that calculation I'm not taking account of the way the sun's g reduces as you approach the center. This will make the orbit take rather longer than our rough calculation.
So - needs a much more complex calculation than I can do easily. Would need also to take account of varying density of the sun, so - I expect it would take more than our hour and a half.
At any rate though, seems to be of the order of hours rather than days. That's enough for now.
Temperature at center of the sun: 15,700,000 degrees Celsius
If our heat sink is a steel sphere (say) it has a specific heat capacity of 0.49 KJ per Kg per degree K and a density of 8,050 kg/m3.
So if it starts off at zero degrees Kelvin and warms up to 20C - just a rough calculation, I know the specific heat varies by temperature, but very approximately we have
0.49*8,050*283 = 1116293 kJ per m³.
So now need to calculate how much heat would be received from the sun during the transition - depending on the size of our heat sink. And how big the heat sink needs to be.
So - here I'm ignoring our passengers for now. They can be positioned inside the heat sink, makes most sense. So you'd need to take account of the volume of the living quarters as well, but I'm expecting the heat sink to be quite big, so that might not make much difference to the calculation.
So now just need to work out how much heat would be absorbed by our sphere, of radius say r meters (or kilometers or whatever).
Might as well assume it is a black body, I doubt if making it a reflector will help since the sun is in direct contact with the outside of our spaceship.
So how much heat is absorbed by a black body of radius r kms, initially at 0K traveling through material at a temperature of 15,700,000 degrees Celsius?
That's the main thing left. Already got everything we need to work out its heat capacity in kJ depending on its radius.
So, we need the Stefan Boltzman law.
This gives the heat emitted by a black body at constant temperature - which is also the heat absorbed by it. Stefan-Boltzmann Law
And the answer is, s*T^4 where T is the absolute temperature, here 25,000,000, and s is 5.6703*10^-8. And result is in joules per m^2 per second.
Which I make 2.2149609*10^22 Joules per m² per second absorbed.
Our Mercury sized "spacesuit" is beginning to look a bit small. But let's do the proper calculation.
Supposing say two hours for our transit (that is the main unknown here as it needs a more detailed calculation and using density variation of the sun to work out the varying gravity)
That's 7200 secs. So our heat sink, has to absorb a total of
1.5947718*10^26 Joules per m².
And it's capacity, assuming it is made of carbon steel, is 1,116293 * 10^6 kJ per m³
So - simple calculation now to work out how big it has to be to do this.
So, 1.59*10^26*4* Πr² = 1.12 10^6 *(4/3)* Πr³
So, r = 1.59*10^26*3/1.12 * 10^6 meters
= 4.3*10^20 meters
or 4.3*10^17 km
Or 4.3*10^17/695842 times the radius of the sun
Or, about 610,000,000,000 times larger than the sun.
So in other words the sun could easily melt a sphere of steel far larger than the sun, in the couple of hours of our sun traverse. That's because it is so amazingly hot.
I've over estimated a bit there because the surface of the sun isn't so hot, but it is hardly worth doing a more detailed calculation when it is nearly twelve orders of magnitude larger than the sun.
This is the idea that, first, the outside of our shielding can melt and reach far higher temperatures than the temperature of the interior - with a layered construction, alternating shielding with insulation.
Platinum melts at 2041 K. So we have nearly an extra power of ten there.
0.49*8,050*2041 = 8050724 kJ per m³.
But after it reaches that temperature, we can't contain it any more. So it would just flow off into the sun.
Then, ablative shielding for Atmospheric entry works by creating a gas layer interface. Could that work for the sun?
I.e. - that our spaceship moving through the sun creates a shock layer - and the shock layer disconnects the heat outside from the heat inside?
Of course we can reduce the transit time by traveling through the sun more quickly. If you can find a way to hit the sun and go through it without heating up instantly. What if you are traveling at close to the speed of light before you hit the sun - and then somehow keep going through it? You could pass through it in seconds, at zero g what's more.
At first sight, that's not going to help that much, because even if we can do the transit in 1 second, the heat of the sun can still melt a steel sphere far larger than the sun.
But - we do also have to consider it as a heat flow problem. With such a fast transition, then the heat mightn't have time to get into the interior of the spaceship before we are out of the other side of the sun.
But I don't think that either of these ideas are really worth exploring. Because...
We have the even bigger issue here, of the density and pressure of the sun.
Been ignoring that so far. But the mean density of the sun is 1,410 kg / m³. Somewhat denser than water. Which might not be too bad if you were traveling at a few knots, like a submarine. But your spaceship has to travel 695842 km in an hour or two, to keep the astronauts at reasonable gravity levels. So you are traveling through it at about 300,000 km / hour average, or around 100 km/ sec.
At the center it has a density of around 150,000 kg/m³, or 150kg/liter which is more than seven times more than platinum at 21,400 kg / m³. Density of Gold, Silver, Platinum
So - flying through the sun - even though it is a gas and less dense than the Earth - would be similar to traveling through the sea for much of the way - and harder than flying through solid platinum at the center, but at speeds of 100 km / second or faster.
Need some pretty amazing "motor" and shielding to do that at speeds fast enough to keep your astronauts at zero g or low g. Even if it wasn't hot at all.
But the pressure issue is even worse. The pressure of the centre of the sun is 233 billion times that of the Earth's atmosphere at sea level.
NASA's Cosmos
By comparison, the deepest ocean depths - at a depth of around 11,000 meters, then the external pressure is roughly 1100 atmospheres. Mariana Trench: The Deepest Depths
So the pressure at the center of the sun is nearly a billion times that of the deepest ocean depths.
So - I think for it to be feasible at all, we need some science fiction materials.
Maybe solid "Neutronium" - material made of solid neutrons - in modern physics we can talk about the centre of a neutron star as "neutronium" but it is not expected to be stable under ordinary temperatures and pressures.
But supposing some kind of strange matter / neutronium could be made stable at room temperature?
Maybe it would have a very high specific heat?
Or maybe, a "neutronium sphere" like that, as dense as a neutron star but stable at ordinary pressure - if it could exist - it would be able to reflect not just radiation but also the bombarding atoms so they have almost no effect on its temperature, maybe can have something that truly is a mirror? And so strong it can resist the internal pressure of the sun?
For something as dense as that, it wouldn't matter that the sun is 7 times denser than platinum at the center. It could travel through the sun without noticing it. Likewise it could travel through the Earth, through planets, through anything made of normal matter as easily as our planes travel through the air. It would need powerful motors, but if you have technology able to create and work with neutronium (if it is possible at all of course), you can probably arrange to have motors powerful enough to travel through ordinary matter without noticing it.
Or a "force field" some way of distorting the structure of space and time. The idea there is - that if you can do that - maybe you can arrange it so that as you pass through the sun, you are insulated from it in some kind of a "space time pocket".
Or another fun idea - if you had the Time Lords "Smaller on the outside" technology, you could have a Tardis with an exterior you can make as small as an atom. Then you could go right through the sun and never notice it.
Tardis "technology" - bigger on the inside, or smaller on the outside.
And here is Clara with a miniature "very much smaller on the outside" Tardis in the recent Flatline.
At any rate it is far, far beyond anything we can consider with present day technology and physics.
However, we could travel through the sun's Corona
It is immensely hot but a near vacuum. Reflection, passive absorption, magnetic field deflection, heat pumps, all those ideas could work in the Corona.
They could even work right down into the Photosphere, the perceived "surface" of the sun. So you could have a spaceship that visually seems to enter the sun as viewed by our telescopes, because at the apparent surface of the sun, it is still a near vacuum.
So, our spacecraft might dip in and out of the photosphere, indeed travel thousands of kms into the "interior" of the sun using such methods to protect it from the heat.
As for the gravity - well it would need to travel fast enough to get out of the sun again without doing any high g turns - and if so - even though the sun's surface gravity is so very high, about 28 gs (The Sun), the inhabitants of the spaceship would experience zero g or low g. That's just like the way inhabitants of the ISS experience zero g or microgravity although traveling around a body with surface gravity of 1 g.
So that's feasible. Sun "skimming" like this seems alogether feasible with not too far distant future technology. You'd have to be supremely confident in its reliability of course.
You can get this and many more of my answers now as a kindle book:
Simple Questions - Surprising Answers - In Astronomy
HEAT - NEEDS REFRIGERATION
First refrigeration. Okay the sun is far too hot for any material we can construct. But what if we are able somehow to dissipate all that heat?
First idea - perfect reflector
If we can somehow reflect the heat away, then our materials will never get hot. So - if you had a perfect reflector, doesn't matter how hot it gets.
But - sadly - we have dense, very hot materials in direct contact with our "heat mirror" - hard to see that working. If we could somehow create a perfect mirror - it could reflect away the heat radiation - but how could it reflect away the heat from the bombarding atoms of the surrounding material?
Second idea, black hole or similar as heat sink
I can think of one way to do it - but it is both extremely dangerous for civilization, and also - hard to imagine it actually working. But suppose we could create a Mini black hole (if such things exist), of substantial size - then it could act as a heat sink because heat going into the black hole would not come back out again. Mini blackholes, if they exist, many physicists think, dissipate due to Hawking radiation. But if you could keep it above that size, just large enough to not produce a significant amount of radiation, then you'd be okay.
I don't recommend this though, what if the black hole escapes from our spaceship, then it is goodbye sun :).
Third idea, just an ordinary heat sink
Or for that matter, doesn't need to be a mini black hole. Make your heat sink out of any super dense material.
Main thing is to start up cold before you go in, and to have enough material in it to absorb all the heat for your transition through the sun.
It doesn't matter if it melts. That would be like the ablative layer on heat shields for re-entry. Indeed if it melts away - and then evaporates - that helps to absorb more heat.
This seems worth looking at closer, see later.
Fourth idea, super dense shell
You could surround yourself with neutron star material. And maybe if it starts off cold, and you go through the sun quickly, you can get through to the other side without heating up significantly.
That is if we can find a way to keep neutron star material in a solid state without it exploding, during the transition. And form it into a shell that you can enter without being turned into spaghetti as you go into the shell.
A shell of super dense material would be okay even if externally it had even several Earth's masses of material and high g from the outside - if you are inside it, you will experience no gravitational force at all. That is also - supposing it is constructed to be perfectly spherical to an extraordinary level of precision. Then you would feel no gravity inside - if you can get inside it in the first place that is.
You could just make it with a hole in the side and you "fall into it" and then rapidly seal the hole. But still, you'd have the tidal forces. As you fall into your protective shelter, at certain times during your fall your feet would be accelerating far more rapidly than your head. And you'd turn into spaghetti. So I think, a bit difficult to "don" this spacesuit, if it is possible at all.
I think what you'd have to do is to build it around you instead. That is if this material is possible at all, more later.
Fifth idea, heat pump
Just for completeness. Don't have to absorb the heat if you can pump it away. I.e. use some other source of power to pump the heat from inside to out using a power source such as electricity. See Thermoelectric cooling - you can cool down spaceships to well below the ambient temperature in space from incoming radiation.
If you could get that to work, and also had an almost unlimited source of power, e.g. fusion power, you could stay in the sun indefinitely, as long as the power source lasts. But - though it works fine for spaceships, could this work in the sun?...
GRAVITY IS NOT A PROBLEM AT ALL - IF YOU HAVE POWERFUL ENGINES TO KEEP YOUR SPACESHIP MOVING
As for the gravity of the sun - if you can shield from the heat - and also - maintain your velocity, you can keep the interior of the spaceship at any desired gravity.
So e.g. if you had a powerful motor programmed to compensate for the resistance of the material you are going through so that it feels like free flight to the residents of the spaceship, you'd be in zero g even on a flight right through the center of the sun. And could come out the other side. For that matter you could also accelerate at 1 g all the way through and come out far faster than you went in, and be in comfort all the way through.
You could even - in a series of orbits, if you can keep the heat away, you could spiral down and end up at the center of the sun and you'd experience no g force there at all of course. You'd be floating in zero g.
FUSION REACTOR TYPE MAGNETIC CONTAINMENT
Fusion reactors - they are close to being able to contain the material for fusion, for long enough for them to fuse, at temperatures far higher than the temperature of the sun. But the problem there is - that it is at very low densities. And also for short time periods.
Could future physics would permit them to hold back material at densities of the center of the sun?
DO WE NEED NEW PHYSICS?
As gravity is not an issue, if we had the "force field" of science fiction, it would be no problem.
Of all the ideas - I think the idea of a heat sink made of some more ordinary material at least is worth exploring. One that is dense enough to absorb all the heat of your transition.
How much heat would it need to absorb? How much mass is needed to absorb all that heat, supposing it starts off at around absolute zero?
Maybe worth doing a bit of a calculation here....
ABLATIVE EXTERIOR LAYER
Note, that we can absorb more heat also if it is designed as an ablative layer so that the outside melts and then evaporates.
Problem there though is - that the outside is far hotter than humans could tolerate. But perhaps could have an external layer that ablates, to get rid of much of the heat in the initial phases - and then an inner layer insulated from it that just absorbs heat - and then the humans inside that.
That way we could get rid of much more heat than we can by just absorbing it in a heat sink.
But I'll ignore that for now, let's see what if we just have an ordinary heat sink, all at same temperature, with humans at centre.
Of course some of the exterior of it will melt and ablate - and that will add to its heat sink capabilities, but not designed as an ablative layer as such all the way through.
GATHERING INFO FOR CALCULATION
Just want some kind of rough back of the envelope calculation here, is it possible at all?
I'm expecting our "spacesuit" to be rather large, kilometers, maybe moon sized, maybe larger. But - it does seem to be at least feasible.
Example. Suppose that we used Mercury as our "spacesuit" with some future technology that permits us to first cool Mercury down to absolute zero, all the way to its core, and then fly Mercury right through the centre of the sun, with rocket motors so powerful we can maintain an artificial "zero g" flight throughout - with the humans inside it?
Surely that would work (without doing the calculation, but seems unlikely that a few hours passage through the sun would make it too hot for humans to tolerate). So - could we make do with something a bit smaller?
Or would it work?
So, let's see.
The sun's radius is 695842 km. So - traveling at zero g, you could get through it in half an orbit of a planet with semi-major diameter of 695842 km.
Using Orbital period of a planet (online calculator) I make the orbital period 2.78043 hours. So you can get through the sun in zero g in about an hour and a half easily.
You'd get through it somewhat more quickly if accelerating at full g all the way. But I don't think that will make a significant difference here since most of the time we are traveling at the full g of the sun.
On the other hand in that calculation I'm not taking account of the way the sun's g reduces as you approach the center. This will make the orbit take rather longer than our rough calculation.
So - needs a much more complex calculation than I can do easily. Would need also to take account of varying density of the sun, so - I expect it would take more than our hour and a half.
At any rate though, seems to be of the order of hours rather than days. That's enough for now.
Temperature at center of the sun: 15,700,000 degrees Celsius
If our heat sink is a steel sphere (say) it has a specific heat capacity of 0.49 KJ per Kg per degree K and a density of 8,050 kg/m3.
So if it starts off at zero degrees Kelvin and warms up to 20C - just a rough calculation, I know the specific heat varies by temperature, but very approximately we have
0.49*8,050*283 = 1116293 kJ per m³.
So now need to calculate how much heat would be received from the sun during the transition - depending on the size of our heat sink. And how big the heat sink needs to be.
So - here I'm ignoring our passengers for now. They can be positioned inside the heat sink, makes most sense. So you'd need to take account of the volume of the living quarters as well, but I'm expecting the heat sink to be quite big, so that might not make much difference to the calculation.
So now just need to work out how much heat would be absorbed by our sphere, of radius say r meters (or kilometers or whatever).
Might as well assume it is a black body, I doubt if making it a reflector will help since the sun is in direct contact with the outside of our spaceship.
So how much heat is absorbed by a black body of radius r kms, initially at 0K traveling through material at a temperature of 15,700,000 degrees Celsius?
That's the main thing left. Already got everything we need to work out its heat capacity in kJ depending on its radius.
So, we need the Stefan Boltzman law.
This gives the heat emitted by a black body at constant temperature - which is also the heat absorbed by it. Stefan-Boltzmann Law
And the answer is, s*T^4 where T is the absolute temperature, here 25,000,000, and s is 5.6703*10^-8. And result is in joules per m^2 per second.
Which I make 2.2149609*10^22 Joules per m² per second absorbed.
Our Mercury sized "spacesuit" is beginning to look a bit small. But let's do the proper calculation.
CALCULATION
Supposing say two hours for our transit (that is the main unknown here as it needs a more detailed calculation and using density variation of the sun to work out the varying gravity)
That's 7200 secs. So our heat sink, has to absorb a total of
1.5947718*10^26 Joules per m².
And it's capacity, assuming it is made of carbon steel, is 1,116293 * 10^6 kJ per m³
So - simple calculation now to work out how big it has to be to do this.
So, 1.59*10^26*4* Πr² = 1.12 10^6 *(4/3)* Πr³
So, r = 1.59*10^26*3/1.12 * 10^6 meters
= 4.3*10^20 meters
or 4.3*10^17 km
Or 4.3*10^17/695842 times the radius of the sun
Or, about 610,000,000,000 times larger than the sun.
So in other words the sun could easily melt a sphere of steel far larger than the sun, in the couple of hours of our sun traverse. That's because it is so amazingly hot.
I've over estimated a bit there because the surface of the sun isn't so hot, but it is hardly worth doing a more detailed calculation when it is nearly twelve orders of magnitude larger than the sun.
ABLATIVE SHIELDING
This is the idea that, first, the outside of our shielding can melt and reach far higher temperatures than the temperature of the interior - with a layered construction, alternating shielding with insulation.
Platinum melts at 2041 K. So we have nearly an extra power of ten there.
0.49*8,050*2041 = 8050724 kJ per m³.
But after it reaches that temperature, we can't contain it any more. So it would just flow off into the sun.
Then, ablative shielding for Atmospheric entry works by creating a gas layer interface. Could that work for the sun?
I.e. - that our spaceship moving through the sun creates a shock layer - and the shock layer disconnects the heat outside from the heat inside?
FASTER TRAVEL
Of course we can reduce the transit time by traveling through the sun more quickly. If you can find a way to hit the sun and go through it without heating up instantly. What if you are traveling at close to the speed of light before you hit the sun - and then somehow keep going through it? You could pass through it in seconds, at zero g what's more.
At first sight, that's not going to help that much, because even if we can do the transit in 1 second, the heat of the sun can still melt a steel sphere far larger than the sun.
But - we do also have to consider it as a heat flow problem. With such a fast transition, then the heat mightn't have time to get into the interior of the spaceship before we are out of the other side of the sun.
But I don't think that either of these ideas are really worth exploring. Because...
DENSITY
We have the even bigger issue here, of the density and pressure of the sun.
Been ignoring that so far. But the mean density of the sun is 1,410 kg / m³. Somewhat denser than water. Which might not be too bad if you were traveling at a few knots, like a submarine. But your spaceship has to travel 695842 km in an hour or two, to keep the astronauts at reasonable gravity levels. So you are traveling through it at about 300,000 km / hour average, or around 100 km/ sec.
At the center it has a density of around 150,000 kg/m³, or 150kg/liter which is more than seven times more than platinum at 21,400 kg / m³. Density of Gold, Silver, Platinum
So - flying through the sun - even though it is a gas and less dense than the Earth - would be similar to traveling through the sea for much of the way - and harder than flying through solid platinum at the center, but at speeds of 100 km / second or faster.
Need some pretty amazing "motor" and shielding to do that at speeds fast enough to keep your astronauts at zero g or low g. Even if it wasn't hot at all.
PRESSURE
But the pressure issue is even worse. The pressure of the centre of the sun is 233 billion times that of the Earth's atmosphere at sea level.
By comparison, the deepest ocean depths - at a depth of around 11,000 meters, then the external pressure is roughly 1100 atmospheres. Mariana Trench: The Deepest Depths
So the pressure at the center of the sun is nearly a billion times that of the deepest ocean depths.
FUTURE PHYSICS
So - I think for it to be feasible at all, we need some science fiction materials.
Maybe solid "Neutronium" - material made of solid neutrons - in modern physics we can talk about the centre of a neutron star as "neutronium" but it is not expected to be stable under ordinary temperatures and pressures.
But supposing some kind of strange matter / neutronium could be made stable at room temperature?
Maybe it would have a very high specific heat?
Or maybe, a "neutronium sphere" like that, as dense as a neutron star but stable at ordinary pressure - if it could exist - it would be able to reflect not just radiation but also the bombarding atoms so they have almost no effect on its temperature, maybe can have something that truly is a mirror? And so strong it can resist the internal pressure of the sun?
For something as dense as that, it wouldn't matter that the sun is 7 times denser than platinum at the center. It could travel through the sun without noticing it. Likewise it could travel through the Earth, through planets, through anything made of normal matter as easily as our planes travel through the air. It would need powerful motors, but if you have technology able to create and work with neutronium (if it is possible at all of course), you can probably arrange to have motors powerful enough to travel through ordinary matter without noticing it.
Or a "force field" some way of distorting the structure of space and time. The idea there is - that if you can do that - maybe you can arrange it so that as you pass through the sun, you are insulated from it in some kind of a "space time pocket".
Or another fun idea - if you had the Time Lords "Smaller on the outside" technology, you could have a Tardis with an exterior you can make as small as an atom. Then you could go right through the sun and never notice it.
Augmented Reality Takes 'Doctor Who' Fans on Inside TARDIS Tour
And here is Clara with a miniature "very much smaller on the outside" Tardis in the recent Flatline.
If the Tardis was made as small on the outside as an atom it would not absorb much of the heat from the sun, just like a light shining into it.
Of course we don't have "Time Lord technology" if such is possible at all :).
At any rate it is far, far beyond anything we can consider with present day technology and physics.
CORONA
However, we could travel through the sun's Corona
It is immensely hot but a near vacuum. Reflection, passive absorption, magnetic field deflection, heat pumps, all those ideas could work in the Corona.
They could even work right down into the Photosphere, the perceived "surface" of the sun. So you could have a spaceship that visually seems to enter the sun as viewed by our telescopes, because at the apparent surface of the sun, it is still a near vacuum.
Higher resolution version here: File:Sun Atmosphere Temperature and Density SkyLab.jpg
By comparison, the Earth's atmosphere is only 1.2 milligrams per cm³. Earth's atmosphere density in 285 measurement units. So even 2,000 kms below the apparent surface of the sun, the sun has about a thousandth of the density of the Earth's atmosphere at sea level.
So, our spacecraft might dip in and out of the photosphere, indeed travel thousands of kms into the "interior" of the sun using such methods to protect it from the heat.
As for the gravity - well it would need to travel fast enough to get out of the sun again without doing any high g turns - and if so - even though the sun's surface gravity is so very high, about 28 gs (The Sun), the inhabitants of the spaceship would experience zero g or low g. That's just like the way inhabitants of the ISS experience zero g or microgravity although traveling around a body with surface gravity of 1 g.
So that's feasible. Sun "skimming" like this seems alogether feasible with not too far distant future technology. You'd have to be supremely confident in its reliability of course.
Earth and Moon on same scale as the sun. In future astronauts could go "sun skimming" even below the surface of the sun in this image.
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About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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