Most countries have signed the UN Convention of the Law of the Sea. (The United States haven't but they largely honor it nonetheless.) This convention defines various types of sea zones, which I won't go into here; Part 7 applies to the high seas, i.e. the areas that have no connection with any state.
The convention specifies rights and obligations related to navigation, including a duty to obey navigation rules to prevent collisions (art. 94, 97) and a duty to render assistance to persons in distress (art. 98). Some activities are forbidden, and any state is entitled to put a stop to them regardless of the nationality of the offender, in particular:
- Transportation of slaves (art. 99)
- Piracy (art. 100–107)
- Drug trafficking (art. 108)
Every ship must fly the flag of one state (art. 91, 92) and is subject to the laws of that state (art. 94). Ships may be considered without nationality if they do not fly a valid flag; such ships can be boarded by warships and government ships of any state (art. 110). While ships without nationality are not forbidden per se, they would have no recourse against the action of a state except within that state's laws.
Owning a private ship is routine and does not preclude your ship from being claimed by your state, and thus all of your state's laws from applying. If you want to sail a ship without nationality, your first hurdle will be to make your state relinquish its claim on the ship — the normal route for that being to register it in another state. Merely omitting registration formalities and not flying the flag might cause you to forfeit the protection that it would bring but will not alleviate the constraints of the laws of your state.
If you manage to obtain a ship without nationality, then apart from a few prohibited activities as mentioned above you can do whatever you like on it. You can murder, but not trade drugs. You'll have essentially no protection, so your ability to survive a life of crime will rest on your ability to hide, or on being too unimportant for anyone to bother.
Your second legal hurdle will be the laws that apply to you as a person. This Straight Dope article has a good overview of that; I'll mostly be summarizing it. Many laws apply both to acts committed on the state's territory and to acts committed by nationals of that state. If you were thinking of violating US laws while located abroad (high seas or not), make sure that that particular law is only in effect on US territory and does not apply to all citizens.
Your third legal hurdle will be the dual situation: states often claim jurisdiction when one of their nationals is injured by an act that would be a crime on their territory. For example, if you murder someone on the high seas, the victim's state will in many cases be entitled to prosecute you. In the United States, 18 USC §7.7 lists “any place outside the jurisdiction of any nation with respect to an offense by or against a national of the United States” as one of the cases where “special maritime and territorial jurisdiction of the United States” applies. 18 USC §1111 specifies that “Within the special maritime and territorial jurisdiction of the United States,
Whoever is guilty of murder in the first degree shall be punished by death or by imprisonment for life (…)”.
I'd expect murder on the high seas to run afoul of at least one country's jurisdiction with most combinations of ship's nationality and victim's nationality. With gambling, you're probably safe if you manage to escape your country's jurisdiction; in the United States, you'd be outside of any federated state, so I think gambling would be perfectly legal. Sex with minors would probably depend on your nationality and your partner's.
A further hurdle will be to benefit from the crime. Payment for any services had better be cash-only, because you'll have a hard time banking without coming into the jurisdiction of that bank's state. And keep in mind that if a customer doesn't pay, you won't be able to sue them.
Finding a tiny no-name uninhabited island in the 21st century (or even for most of the 20th) is a tall order. It could happen — every now and then a volcanic island is born — but you'd need to be diligent and lucky. Your best chances to find land which is not claim by any state are in fact on continents:
- Marie Byrd Land, a sector of land in Antarctica, is unclaimed. Bring a sweater.
- Bir Tawil, a sliver of land in the eastern Sahara between Egypt and Sudan, is claimed by neither country (they both claim the Hala'ib Triangle instead). Bring plenty of water.
You can try to set up shop in unclaimed territory declare yourself an independent country — a micronation. Artificial islands are a popular choice. You won't be the first. Your laws would apply, but you would be without any state's protection, so it would be up to you to enforce them. Furthermore trade with entities located outside your micronation would be difficult. Most countries would not recognize your passport, so forget about traveling. And, just like a nationless ship, the continuation of your authority would depend on both your ability to resist any internal challenge (including assault by raiders, as there is no international convention on the prevention of piracy on land) and any challenge by a “proper” state who might one day decide to claim your land.
Sorry to burst the bubble of your geek's dreams, but you can't just become a nation by saying so. You need credibility, and that doesn't come cheap. Even having an army and controlling a large amount of land doesn't automatically lead to international recognition — over a billion people live in a state that is not recognized by some other state (admittedly, this doesn't prevent these people from interacting normally with people, corporations and even governments of non-recognizing states). To get governments to talk to you, you'll need to convince them to take interest. Running a business of bypassing their laws might elicit their interest, but probably not in a way that aligns with your objectives.