In the case of Ms. Duran, she says that simply making the claim is therefore sufficient to ensure her ownership of the Sun. In the past, a guy named Dennis Hope made the same claim — he sent letters to the government basically saying that if they don’t reply, they are giving tacit permission for him to claim the Moon. Not surprisingly, the government ignored him, so Hope now says he owns the Moon. He even took it farther, having sold deeds to property on the Moon… and he’s not going broke, having made millions doing this. At least.
However, there is more to international law than the words in the treaty. Plait explains why Duran's claim to the sun will not hold up. Link