The surface of Earth is actually a relatively flat disc accelerating through space "upward" at a rate of 9.8 m/s^2, not a globe. The north pole is at about the center of the disc, while Antarctica is the "pizza crust" on the outside. The rest of the universe is moving and accelerating such that all the observations seen today by amateur astronomers are produced. The true nature of the sun, moon, stars, other planets, etc. is not yet well-understood by science. A conspiracy involving NASA and other space agencies, all astronauts, and probably at least some professional astronomers is a necessary element. I'm pretty confident this isn't true, much more due to the conspiracy element than the astronomy element, but I don't immediately dismiss it where I imagine most LW-ers would, so let's say 1%.

The Flat Earth Society has more on this, if you're interested. It would probably benefit from a typical, interested LW participant. (This belief isn't the FES orthodoxy, but it's heavily based on a spate of discussion I had on the FES forums several years ago.)

Edit: On reflection, 1% is too high. Instead, let's say "Just the barest inkling more plausible than something immediately and rigorously disprovable with household items and a free rainy afternoon."