Legally though you can't own stuff in space, there are already two treaties, the Outer Space Treaty from 1967 and the Moon Treaty from 1979 prohibit such practices. Of course, some countries have signed those treaties and others have not, and there could be legal loopholes (the first one states that no state can own extraterrestrial property, but doesn't say individual cannot, etc.) but you also can't just claim land like that. You've got to have intent to occupy, which is impossible for a private citizen at the time.
I wonder though, if through some insane legal battles you can will your land titles on the Moon to your children, they to their children, and so on and so forth, until at some point it is legal to own land on the Moon (because you can live there), will the laws be a retroactive acknowledgement of the family deeds to 50% of the Moon bought for 3 cents back in the summer of 21.
If that's the case, then I'm buying all the stars in one of the arms of the Milky Way and my future descendants will be rich as they collect the solar tax on those billion stars and Dyson Sphere-ing rights.