Just wanted to add, to these excellent answers, that in a way the world is flat to a first approximation locally. And of course its rare in everyday life that you need to take account of its curvature. You don't need to build with curved foundation blocks :).

Also, we can't combine together in our imagination this world where you can walk for miles and miles over the landscape with the idea of something round.

You can attempt to, but you are bound to think of it as a small round object.

It's so close to flat almost impossible to imagine something as large as our Earth and also round.

Photos from space show the world as a small round sphere. And from that perspective it is. And you see how fragile it is. And nowadays it is possible to fly around the Earth in a couple of days if you travel non stop. How long would it take to go around the world in a plane, nonstop?

But in another sense it is so vast - circumference of 40,075 km.

At a typical walking speed of say, 5 km / hour, that's 334 days, or a little short of a year to walk all the way around, walking day and night. If you walk for only 8 hours a day and keep up that 5 km, so traveling 40 km every day, or 25 miles a day, you would take three years to walk that distance.

Of course much of it is also ocean. 

But I think the flat Earth society - though obviously wrong from a scientific point of view, do have a point psychologically and perhaps philosophically.

To get an adequate picture of the Earth we need to combine, somehow, the vast perspective you get with the idea of a flat Earth stretching as far as you can see in any direction, with the round Earth picture of science. Easy to combine in science and maths, not so easy in imagination.

OF COURSE YOU CAN TELL IT IS ROUND

Just to say, don't get me wrong, I'm not saying you can't tell that the Earth is round :).

If you go to the sea shore especially, you can probably tell that the distance you can see depends on the height of your eye above the horizon. An island for instance may be hidden when you look from 2 meters above sea level but visible if you climb a small hill, and you can see boats disappear below the horizon as they depart on a calm day. Distance to the Horizon Calculator 

And there are many other ways you can tell it is round mentioned in the other answers here. For instance by looking at the shadow of the Earth on the Moon, or by comparing shadows of vertical sticks at different latitudes.

I haven't gone into them in my answer because they have been said already. See the other answers to Let's say I don't believe the world is round. How can one prove the world is round to me?

But locally it is extremely flat, so much so that until the Greeks in the 5th century BC, just about everyone thought it was flat and all the ancient civilizations before then thought that it was flat. And in India and China and other places the idea of a flat Earth persisted into recent times, though it's a myth that Europeans thought the Earth was flat before Columbus. For more on this see the other answers here + the wikipedia page . Flat Earth

And that I think if you have got it into your mind that the Earth is round, you may find you think of it as also really small. In a way it is, relative to the sun, planets, scale of the solar system. But in other ways it is not. It's absolutely huge. Especially if you feel that somehow we are confined into a tiny planet, that may be because you are thinking of the Earth as far smaller than it really is.

It is both fragile, and also vast beyond imagination. And it isn't as fragile as many think now. See for instance Will Earth ever be as inhospitable to humans as Mars is now?. It is both interconnected, and fragile - but also robust as well in other ways.

I think many people when they imagine a round Earth think of something a bit like this

And that view of the Earth is just as wrong as the idea that it is flat. :).

Or, a better way of looking at it, both ways of thinking of the Earth have an element of truth in them. Not scientific truth, there is no scientific truth in the idea of a flat Earth, except as a useful first approximation for many experiments where you don't have to take account of the curvature of the Earth.

Even in science it is usually a good approximation to neglect the roundness of the Earth or the curvature of space time it causes.

But though it's not a scientific truth except as an approximation - there's an element of philosophical and psychological truth.

I come to this from a slightly different perspective, don't often encounter people who are convinced the Earth is flat, but often come across people who think the world is doomed by our technology and I think that the idea of the Earth as a tiny round sphere which sort of pervades our culture may be part of that, that they don't realize quite how huge it is. And how resilient it is as well.

Even with our modern technology and vast numbers of people, there is nothing we are doing that will severely impact on the habitability of the Earth. Will Earth ever be as inhospitable to humans as Mars is now? When you accept that, then you can start to work on the things that do need to be dealt with.

I think this idea of the Earth as so fragile is something that can be paralysing if taken too far - like the two extremes - that Earth is so robust nothing we can do will impact on it - obviously wrong. And that it is so fragile that we risk making it uninhabitable - that's also wrong. We are in an in between state where things we are doing do have impacts on the Earth. But it is also  resilient if we can just find a way to work with it, to help it heal itself. Which need not be that hard to do.

You can try it out, just try for a moment or two imagining that the world is actually flat rather than round, stretching endlessly in all directions - and see how much larger it seems to be in your imagination than your picture of  round Earth.