Is everything in outer space hefker? For example if a private person lands on the moon or an asteroid can he be koneh it? Do you retain the rights to something that you send into space?
First you have to differentiate between Reshut HaRabim and Hefker.
Nobody can do a Kinyan on Reshut haRabim; it belongs to everybody unless the "king" or local equivalent decides otherwise. (Unless you cause damage; it belongs to you insofar as you are responsible for restitution.)
When a tract of land is Hefker you can do a Kinyan - appropriate to land like digging in it - but only that piece of land you dug up is yours. Unless it's fenced in, but then it's probably not Hefker. Fencing in a tract of land may be a Kinyan.
(Based on my reading Tur and Shulchan Aruch Choshen HaMishpat these past few months. How to do Kinyan on an heirless Ger's estate comes to mind.)
So the question really becomes: do things in outer space have the status of Hefker or Reshut haRabim.
As was pointed out, you may not be able to claim ownership of the sea floor, as it probably is Reshut haRabim by common agreement. Once out of territorial waters, ships seem to have a "free route" and nobody would want to change that.
So, do we (sensible earthlings) want outer space to be Hefker or Reshut haRabim?