Her argument also hinges on the belief that a notary in Spain somehow has the authority to validate her claim to ownership of the Sun. This is absurd for a very simple reason: A nation or government can only exercise sovereignty over the people and territory within its borders or of its own citizens while outside its borders. The sun does not lie within the territorial jurisdiction of Spain or any other country, nor is it subject to the authority of any government or other organization.
It's the same with the Moon example in the article. The moon has no government or permanent residents. No entity has established sovereignty over the moon. Therefore nobody can "own" the moon. Now if you flew there and set up a colony, then you could plausibly claim that somebody "owns" the moon.