Lots of people think they own a piece of the moon. You yourself can buy a plot of lunar land on ebay if you want. The problem is, there is some confusion over who actually owns the whole moon and therefore has the right to sell it (or bits of it). One guy says he laid claim to it first, another says he did. Another area of debate is that even if the guy who claims to own it is granted official ownership by the US government, who's to say the US government has the right to say 'yes, that guy legally owns it'.

Bottom line: Nobody owns the moon.

But if somebody officially did, and it was agreed the world over that he did, I would definitely buy a plot from him - and when we start colonising, I (or my kids) will make a killing.