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?
You couldn't do it halachically anyway. Treaties signed by the major nations declare space as the common heritage of mankind, with ownership of any portion of celestial body by any nation explicitly forbidden. Most opinion articles seem to believe this also extends to private ownership, that the goods created by any private party must be equally shared among all the people of Earth, rich and poor nations alike.
If you are a citizen of a country who is signatory to these treaties, then as a Jew you must obey that treaty, since that is the law of your land. Therefore, sanctification of ownership in this instance should be meaningless.
To answer this question you have to understand the idea of the property on Earth. All monetary matters are subject to social approvement, in other words, "דינא דמלכותא דינא".
So imagine you find an iceberg in Iceland and you'd like to privatize it - is it Halachicly yours? The only way to answer this is to ask others - It makes no sense of having a property with no humans around, think about that.
So there is no difference in boundaries for that social approval - you might claim you own 5 square miles on Uranus, but as long as there's no existing agreement about Uranus, your claim is meaningless.