The observational and scientific data to date has disproven flat earth theory, but more interesting I think, indicates that the earth is at the center of the universe, which is another matter entirely. Anyways, we can pretty much say that the earth is a sphere and that there is space because we DID land on the moon. Russia was pissed that we won that race. If the earth was flat, they wouldn't have admitted anything, but kept silent instead. Also, we know that the moon landing was not filmed in a desert because faking it in 1969 would have been much, much more difficult and expensive, and pretty much impossible, than just going to the moon. It's all in the way the shadows were cast when looking at the photographs and film - they were parallel. They were cast by sun rays with no diffusion - not divergent as you would get on a film stage. Using studio lightning, which is a nearby point source, rather than 93 million miles away, would cause it's shadows to diverge. We simply didn't have the lighting tech nor the film-making capability at the time to fake it, which at the time would have required millions of lasers, far more than existed at the time.

Secondly, the only lasers to exist then were very large and only created red light, and they were expensive, and even attempting to assemble any apparatus of millions of large lasers would have far, far exceeded the cost of the entire Apollo project. It would have exceeded the budget of the United States itself many, many times over. In addition to Russia, scientists from Australia, Spain, and England at the time of the landing all independently picked up Apollo's transmissions from space and verified Apollo's positions and trajectories. Why would so many governments lie on our behalf to protect us when they were jealous that we got there first?

And if that isn't enough proof for you, Apollo also left behind a retro reflector on the moon so anyone can point a laser at it and get a return signal bounced right back. If the earth were flat, none of this would be possible.