The concept of territorial water off the coast of nations was originally set based on the maximum range of shore-based cannon (three nautical miles). You could claim it and enforce it. Now there are all kinds of international agreements and rules, etc. National airspace was enforceable when airplanes were first invented because you could shoot them down. When satellites were first put up, only the countries that could put them there had the capability of removing them, so countries that wanted to claim sovereignty out into space just had to sit and grumble. Now, more countries might have the capability of shooting down satellites, but the countries that launch satellites and want space to be free for use still have the greatest military capability.
Just to complicate things, eventually, we will be able to mine the moon and asteroids, etc. and there will have to be some means of establishing ownership of these minerals. There is a private group that is planning to sent a (unmanned) lander to the moon sometime in the next few years. As private space capability increases, we may find some very thorny issues have to be worked out.
Now the left half of the Moon on the other hand is mine! I've staked a claim and put up a claim stake on it and I challenge any of you to go there and prove me wrong!