Sure, you could buy a piece of the moon, and it'd be legally yours... at least until someone actually lands in your spot and sets up shop. It worked the exact same way back in the late 16th-early 18th centuries with land in North America (and again in the 19th century when the US was in its expansion era). A person could easily buy a large amount of land on the Ohio River, but unless they could back it up, their deed was worth about as much as the paper it was written on.
Long story short: if you buy a piece of the moon, be prepared to build your own rocket to get there and back it up.
Reminds me of a friend who had three Plutos FedEx'd to him overnight...