What happens if a new volcanic island forms in the middle of the ocean, outside of any nation's borders? Could someone theoretically claim it and start their own sovereign nation?
Indeed, but most of it is very deep and won't be seeing any new islands soon.
Also see wikipedia "principality of sealand". Some british nut tested a lot of the legal precedents. Short version, it totally works, although there was that one time his nation was conquered by mercenaries hired by a german lawyer, and his son was taken hostage. So, self defense is a very real concern.
Given that, there is no benefit to leaving exclusion zone. You can just set up in international waters much closer to shore.