I found a website run by an interesting fellow called 'Wild Heretic' and it seems incredibly intricate and comprehensive. I've yet to see any other person argue as well for half so radical a claim. Think of this as an opportunity to examine arguments for highly unpopular views.
Wild Heretic believes that we live on the inside of a hollow sphere, lit by a half-light half-dark Sun at its center (he claims that light bends in order to produce the effect of rising and setting), that the moon is an optical illusion, that manmade satellites don't really exist, that the stars are light artifacts produced in the atmosphere and can never be seen above it, and he has a bunch of explanations for the other celestial bodies like comets and galaxies.
It all seems shockingly intelligent (aside from when he insists that the fact that the Earth doesn't move under your feet when you jump disproves heliocentrism). He also has nine main pieces of evidence for his model:
1. Some early modern maps have inversed latitude and longitude
2. Modern polyconic maps show more accurate sizes and shapes
3. 19th century balloon observations (that is, without an intervening medium) gave the impression of a concave surface
4. 4,000 foot plumb lines reportedly were farther away from each other at the bottom of a mine shaft
5. A laser shot between two posts (over water) seems to curve downwards
6. An old rectilineator experiment indicates a concave surface (the experiment has been criticized here)
7. Radar and radio wave horizons cannot be explained on a convex ball
8. Ships disappearing below the horizon are an optical illusion
9. Light bends upwards, which allows for the rising/setting illusion of the sun and moon
I would really like to know what people here have to say about this, since the comments on the site itself are very disappointing. (A lot of it does rely on a massive conspiracy involving scientists of many stripes, but it's probably best to overlook that.)