Law of the sea is a bit complicated, but if you want to control your own laws and stuff on an island (OR an old oil rig or whatever), it goes like this:
24 miles out: most likely, the owner of the nearest coast can exert its laws over you for all kinds of things like taxation and blah blah we don't want.
25-200 miles out: the nearest country cannot exert legal force over you for your lifestyle or general activities, but they do have exclusive economic rights, so you can't go fishing or drill oil or gather rocks or anything. Basically makes it impossible to survive. I guess maybe you could live off of hydroponics or something.
200-350 miles: if the continental shelf extends beyond 200 miles, then up to the edge of the continental shelf, the nearest nation still has economic rights to minerals in the sea bed, but not fishing. Max range on this is 350 miles, so even if the shelf goes further, all rights stop there. It might be difficult to get official nation recognition, but in the meantime, no nation would have a legitimate claim to control or regulate any of your activities. So de facto you are a nation.
I am not sure if any islands actually exists 200+ miles away from any claimed territory by an existing nation. And it'd be almost impossible to build an artificial island outside of a continental shelf.
The best bet (assuming no islands exist, I could be wrong), would be to find some place where the continental shelf goes further than 350 miles, and build an artificial island still on the shelf just outside of that range.
OR
Just get any old island and do whatever you want, then militantly defend its sovereignty until some nations acknowledge your legitimacy. Most importantly the one whose zone you are in. High chance of injury or death or imprisonment.