Apparently there was an artist who wore an eyepatch for a few months, after which his brain stopped the 3d effect, and he essentially was seeing in 2d.
This of course was all stuff told to me in an art class on perspective, so take with a grain of salt.
Don't take my word for it, close one eye and move your head from side to side and see the 3d jump out at you.
Speaking as someone who naturally has little to no depth perception (amblyopia affecting my right eye), I still see 3D just fine, catch flying objects (balls, frisbees, etc.) just fine, and am not disabled in any significant way except an inability to see into those magic eye puzzles.
Two eyes give you the ability to detect parallax without moving your head; those of us without binocular vision detect parallax by slight (imperceptible) movements of our head. There are many other useful visual artifacts of our 3D world: closer things are larger than farther things, closer things obscure farther things, and so on. Our brains just construct 3D worlds based on those data rather than the simpler, more reliable stationary parallax detection that you two-eyed folks use.
But the difference between what each eye sees is also an important factor in 3D vision. Movies just make it the only factor, which is probably why they can be disorienting. My guess would be that it's more important for things that are close than things that are far away; something 3 inches from your face appears to be in drastically different positions if you cover alternate eyes, but distant mountains look about the same.
Know what's really hard? Grab a flashing/blinking bouncy ball, and try to play catch with it (or juggle) in the dark.
This is particularly striking since batting involves judging the ball that comes at you with some speed. His previous judgement and muscle memory might be useful of course. But I'd love to read about any research on this, if anyone is aware...
Unfortunately, despite this, they weren't willing to let him go back to flying fighter planes, which is why he got into space suit research...
They do it by moving their head - watch for a while, they almost always run a little bit along the branch before jumping. Sort of like synthetic aperture radar.