OK; so to start a new country today without leaving the planet, we have to turn to the high seas outside of any international boundary. In essence, a group would have to come up with the finances to build a floating island in the mid Pacific or Atlantic. It would have to be moored to an underwater mount to prevent drift into any sovereign territorial waters and the risk of being taken over. There are ideas out there concerning this, particularly by the Venus Project, which by the way, I do not support.
If enough people get together and build this floating island, it could then declare itself a sovereign and independent state, draft its own constitution, elect its own government if so choosing, create its own currency, grow its own food supply on the island in vertical farms, farm fish, desalinate water and provide its own power by a number of ways. We have the technology right now to do this. All that is needed is the resolve.
But the state of the world right now is one of power, conquest and imperialist adventurism and expansion, so another consideration might have to be some form of self defense. These are some of the solutions and some of the problems.