For the moon and everything off the planet there are international agreements that none can be owned by any country. Same for people, I imagine.
Nope. The Outer Space Treaty only forbids national claims to territory (or more specifically, "national appropriation"), since in the context of the 1960s there wasn't a lot of reason to get into personal or corporate claims. Even then, there's questions about how significant a ban this actually is, since you could easily put forward reasonable claims that avoid explicitly saying that you're appropriating territories but are, in effect, equivalent. For instance, if you built a base on the Moon, then most people would agree that you could forbid other people from acting in a certain radius of the base without your permission, to avoid accidental damage or injury from crashes. But in effect this gives you a small territory that you control, thereby bypassing the Outer Space Treaty.
The limits of the Outer Space Treaty are probably the hottest point of debate currently in space law. No one's quite sure precisely what it actually limits and what it permits.
Also stuff in Antartica about freezing any land claims and loads of countries have bases dotting the area. Argentina seems to be the only ones to much care about their claim though.
And Chile, don't forget Chile.