It’s not impossible, of course. As you point out, nations do it.
So theoretically and on that basis, so could you. That’s what makes it impractical. You’re not about to wrest any part of Missouri away from the United States to claim sovereign status (been tried before, as you know, and on a much larger scale—disastrous results re-enacted yearly). So that leaves other and less-powerful or less mindful nations where you could “sneak” some property away from their control.
Good luck finding that place.
Alternatively, you may yet discover the as-yet-undiscovered island in the middle of the ocean somewhere and claim it. Note that it would have to be outside of the territorial limits of any extant government (which makes the South China Sea unlikely, for example, as well as much of the rest of the South Pacific).
Then there’s always the time-honored national custom of just taking it from whoever already owns it. The problems that this tack presents are not insignificant: The United Nations tends to frown on this, and unless you have a supply of atomic weaponry and delivery systems and the means to protect them (or a Dr. No type death ray or equivalent), then a relatively small force would re-establish the “rightful owner” of the place to them.
It may be possible to purchase sovereign rights from an existing government, or to buy such a government outright. After all, that’s pretty much how the Sultan of Brunei rules, isn’t it?
If there’s blood involved, then you really have to watch out for the UN. And if you’re a US citizen I think you’re pretty much S.O.L. anyway, because as far as I know you’d have to renounce your US citizenship in order to be a sovereign somewhere else.