In lakes the rules are extremely strict, for an island in the ocean the rules are much more permissive. No matter what the rules you typically want to keep your impact as low as possible or your island feeling will soon be replaced by something more akin to living in a small polluted town.

Legally all your island really is is a bit of land surrounded by shoreline and that shoreline is the interesting bit. It may automatically be public access or not depending on where you are, you may be accidentally part of some incorporated town and there may be specific rules governing islands.

I would advice against issuing your own passports but if you happen to have an anti-aircraft gun and an old torpedo lying around you could even consider going sovereign.

And once you successfully pull that off you can do whatever you want with your island ;)