Eddie Izzard might have you believe that it all comes down to a cunning use of flags. In truth, the USA only owns the junk they left up there. The Moon falls under the protection of international law as the “common heritage of mankind”. In other words:
Outer space, including the moon and other celestial bodies, is not subject to national appropriation by claim of sovereignty, by means of use or occupation, or by any other means’.
It is an equitable agreement. The good news is that if you have the money, power and connections, then are pretty much free to start homesteading the Moon for your own benefit. Private organizations can also lay claim to the Moon, more likely if they have revenues that can dwarf GDPs. What if in their hubris they end up mining the Moon and awaken an army of space Nazis!?!?!?! It’s not a far fetched thought. Check this out.
In any case, the complexity and logistics involved will drive you slightly bonkers. For now, take Vsauce’s word and don’t start buying orbits and lunar parking spaces. Watch the video to learn all about international space treaties, defending the Moon and getting away with extra-terrestrial murder. Don’t be discouraged if all of this seems like an impossible fantasy. There is nothing keeping you from picking your nose clean for that precious Moon dust 😉