Gah, I just wrote a big long post and it didn't save ><
As to the question of who owns Antarctica, it sort of boils down to this: In 1959 there were 12 nations who all held a claim to a portion of the territory. They got together and signed a treaty stating the land belongs to all of them and that it must be used for peaceful research. But the weird thing about the treaty is it didn't actually require any of the nations to relinquish their existing territorial claims. So it's sort of Schrodinger's Continent, both owned and not owned, depending on how you look at it. But that doesn't seem to have stopped other countries who were not signatories to the original treaty from building bases. China is apparently all over the place down there.
However, from my initial Googling, it does seem like there is an updated version of the treaty and only nations that have signed on are allowed to really be there, at least officially. I think if we were to do this the right way we'd need Asgardia to apply to join the treaty, and I'm guessing we'd first need to be recognized by the UN, so we're looking at piles of legal and diplomatic hurdles.
Unless, of course, we go rogue and just sort of... show up. But the question is, would we want to? For me it comes down to a cost/benefit analysis. It would take a significant amount of resources to build a base there, and I'm not sure what type of research we would be doing or what we'd hope to gain from it. So for me, I don't know that it's worth it at this point. When it comes to colonizing and nation building, apart from the diplomatic hurdles, the climate challenges alone are so forbidding that I don't know if it's a wise move. If something were to go terribly wrong, apart from the human tragedy, it would be a PR nightmare that our fledgling nation may not recover from.