Let's say one day, as I sail across the pacific, I come across an uncharted island.
This island is not anywhere near the borders of existing countries. It is unoccupied.
Can I put up a flag on the island, state I own this piece of land, and run my own country on top of it?
Technically you could stake such a claim in accordance with general principles of public international law, but every country has discretion about whether to recognize you as a sovereign state. Established states would likely have no interest in dealing with you until you have a certain amount of influence and resources. You might have to defend your claimed land and property by force at times from pirates, imperialists, states claiming authority by international agreement, etc. There are a number of microstates and micronations with varying degrees of recognition.
International law is a tricky subject, because technically, the world is in anarchy, not all countries agree on everything (which is the number one reason countries exist).
If there's one thing I learned in my political science classes in University, it's that coersion enforces the law, and the law is whatever those with the powers of coersion say it is. If you can take the island, it's yours, if you can fight to keep it, it remains yours.
Yes you can have whater island you want, but you also need an army in order to keep it, whether it be your own personal army, or a contracted army (private or government). Otherwise, what's to stop a band of pirates, or nearby state from taking your island after you've built up a nice infastructure?
You also need allies. Even if country "A" says that you can go ahead and claim an island, country "B" may have a different oppinion, but despite their oppinion, they may stay their forces from running you off the island if they know country "A" has your back.
Before you go stake your claim on some uninhabited beach, I'd suggest you become familiar with what kind of policies your political neigbours might have, since they are most likely the ones you're going to have to deal with if there are any disputes over your claim of sovereignty on the island.
