First, let's imagine a whiteboard. This whiteboard will serve as our field of vision. Anything on the whiteboard we can "see," and anything not on the whiteboard is outside of our field of vision. Say this whiteboard is 10m long, and 3m high. Pretty big whiteboard, plenty big enough to encompass our whole field of vision, even if we stand a foot or two away.
Now, let's draw a circle on this whiteboard. Let's start it at a 1m radius. This circle neatly fits in the center of the whiteboard, and we can clearly see that it is, in fact, a circle. Now, let's make that circle a little bit larger. Let's make it's radius 2m (we will draw this so that the bottom of the circle is cut off, and we can see the top of it). Some of this circle is no longer in our field of vision, but the top is still curved enough that we can still see that it is a circle.
Now for a jump: let's make the circle a radius of 5m. Once again, we will draw this so that the top of the curve is visible. Now, most of the bottom is cut off. The circle looks more like a bump, from what we can see. But it is still no stretch of the imagination that this is still, in fact, a circle.
Another jump: A circle with a radius of 20m. Now all we can see is a gentle curve, rising less than 2 m from endpoint to endpoint. Still, we can presume from the visible curve that it is a circle.
Yet another jump: A circle with a radius of 100m. The curve remaining inside the whiteboard is scarcely visible. One must draw a line from endpoint to endpoint on the whiteboard to see it very clearly. It almost looks like a straight line.
One more jump: A circle with a radius of 1km. No curve is readily visible. Only with a straight line from endpoint to endpoint can we perceive the tiniest of deviations from it.
This is the time I remind you that this whiteboard is 10m in length. Pretty long whiteboard.
I lied, there's another jump: Finally, a circle with a radius of 500km. Even with the straight line from endpoint to endpoint, there is no repeat no perceptible difference. One more time: it is impossible to see any difference between the straight line and the curve. Yet we are still at 1/30th of the radius of the Earth.
The reason it seems that the earth is flat and that there is no curvature to it is that the radius of the "circle" we are observing is so large. We can't see all of it, and the part we can see curves so little, it's impossible to detect.
Also, astronauts.