Okay - as far as I know, the heaviest spacecraft ever designed, with the idea that it might possibly fly some day, is the interstellar version of Project Orion.
This is the interplanetary version, a mere "few thousand tons"
The largest interstellar one has a diameter of 20 km, and a weight of 10 million tons. So - your aircraft carrier could fit across its diameter (not length) 30 times over.
Theoretically it could reach getting on for 10% of the speed of light and is one of the few practical methods we have for interstellar travel.
It was projected to cost 1 year of the US GNP in 1968, and in 1968 dollars, a little short of 4 trillion dollars.
That is an enormous amount, but also - surprisingly low for such a huge rocket.
The reason was - that it was going to be powered by exploding nuclear bombs, one after another, with the bombs exploaded every 100 seconds.
It was eventually halted - but not for what might seem the most obvious reason - hazard to those on board. Because - turns out - that it is actually reasonably safe - so long as you have a nice thick "pusher plate".
It is the only spacecraft design I know of where it is actually a benefit for the spacecraft to be really heavy - to have something for the bombs to explode against and push forward.
And - it would have been able to take off from the Earth.
The big problem of course is fall-out.
So - you wouldn't even consider it now but back then when Freeman Dyson invented this, we didn't know so much about fallout. For a year or two, the US government was seriously considering this, along with its other peaceful nuclear weapons ideas.
It was stopped eventually when they realised it would cause increased risk of cancer for population around the launch site, for every launch.
We could still build it in space though - if we can find some source of uranium or some such in space - and find a way to refine plutonium in space - except - that it might fall foul of the Outer Space Treaty.
I'm not totally sure if it would.
We certainly can't launch it from the Earth's surface as that would break the Nuclear Test Ban treaty.
But if built in space and launched in space - and if only used for peaceful use - does this break the OST?
The OST doesn't specifically mention nuclear weapons or bombs. It talks about "weapons of mass destruction"
Are nuclear bombs used solely as a method of propulsion a "weapon of mass destruction"?
I imagine lawyers might discuss that for some time!
Either way - there are no plans to build it at present, and not likely in near future.
But you only said "Could be built" not "Would be built". So maybe this applies.
I can imagine some scenarios where it might actually be built.
Suppose we see a huge comet heading our way from the Oort cloud - one large enough to devastate a city or cause a tsunami.
I could imagine if there is no other way to stop it - that we might build one of these. Would have to have some kind of temporary waiving of the test ban treaty - and special understanding for the OST.
But laws are created by humans and nations, and for us. So - in a situation like that - well we could all come together and agree that just for this one off occasion, to permit the Orion to be launched.
And then at day of launch and for some time after, we all have to wear radiation masks, maybe even radiation suits - but it might be thought worthwhile.
In that very special situation. Which is not that likely to happen, tiny fraction of a percent chance of it happening before 2100, an imcoming asteroid so large that you need an Orion spacecraft to stop it.
This by the way should not be confused with the newer Orion spacecraft, which is a different spacecraft idea.
Also - didn't notice (or has the question been changed?) - it also says built in orbit. If built in orbit, using uranium mined in space, then you don't have those radiation problems. Though that is more distant future technology - would be hard enough to build on the surface, require high space technology to build in space.
I'm assuming you mean largest in terms of crew. If you mean largest in physical dimension - well large solar sails would be multiple kilometers across so including the solar sail would probably be the largest spacecraft around, and visible as very bright stars in the sky, maybe even as second moons.
JPL apparently also looked at idea of a spinning ring enforced solar sail with Mars gravity around the outside so carrying crew under mars gravity.
For interstellar propulsion, Robert Forward now suggests "gray sails" mad e of carbon - covered in aluminium. Aluminium used as traditional solar sail to navigate to vicinity of the sun. Aluminium then evaporates as it passes within 3 solar diameters from the suns disk, exposing an underlying layer of carbon that then absorbs sunlight, heats to 2,000 K and re-radiates it as infra-red. This would accelerate the craft at 14 G for as long as it is close to the sun. Still would take a long time to get to another star, like a couple of centuries, but it would zip through the solar system very quickly.
For launch from Earth surface, physically largest would be the orbital airships, of JP Aerospace.
These again would be kilometer scale. The reason being that though hydrogen is still a lifting gas in the near vacuum of the upper atmosphere - to have significant amounts of lift you need a huge airship. But on the other hand it is not subject to the storms and wind damage of the troposphere. So it doesn't need robust construction. And is so high in the atmosphere that equipping the crew module with a parachute - and re-entry aeroshell - is all you need for safety there.