Can I live on an unoccupied island?

Much better to make an artificial island, so that you can move it where you want.
A friend of mine, a journalist, has created one in Amsterdan with recycled PET in early 2000.
Then he covered it with grass, a nice house and some trees.
He also wrote a book about that experiment.

Actually, if you take a piece of land, and no one claims it for a while (meaning enact court action against you on order to recover the legal possession of such a land) you will eventually acquire the property/ownership of it through the legal institute that is called “Acquisitive Prescription” in US and UK law systems (it is called “Usucaption” from Latin “usucapio” in Civil Law systems such as France and Germany).
The institute shouldn’t consider your country of citizenship on regards of your ownership rights arising from acquisitive prescription, however some countries and jurisdictions may do consider it.
In that case only a national of those countries would be able to achieve ownership rights through acquisitive prescription or usucaption.
Just do some research on court cases and decisions on acquisitive prescription (or usucaption, usucapio, or something like, if you looking to move to a country which legal system is based on civil, and not common, law).

Through this research you will find how long you will need to remain as a peaceful occupant of a land parcel to claim its ownership through acquisitive prescription.
Then go there, look for a piece of “res dereclita” (abandoned land) and take it over.
If no one goes against you in a court of law to recover that land, it will eventually be legally yours.

Well, if you live there, by definition it will no longer be unoccupied 😉

Seriously, though; I think there would be no problem legally – if no country or person owns it, I think no-one could stop you living there for free, even if that would mean setting up a tent and catching fish.

The key is “and nobody knows about it” – I doubt such an island exists, much less in the habitable zones of our planet. All the land, including little rocky islands, is known and mapped and claimed.

In some countries, land that nobody holds a deed for belongs to the country itself and you are forbidden to do certain things (such as building houses) on it. See Crown Land for example. It’s open to the public but it’s not yours to do with as you will. In other countries, land belongs (in large swaths including small islands just offshore) to the nearest village, and a visitor can’t so much as swim at a beach without paying a small fee to the village chief. In still other countries, there just isn’t any land that nobody holds a deed for, and that includes your “deserted” islands with nobody appearing to know about them.

The days of “I claim this land for X” are long gone. You would have to buy the island, and there’s a good chance whoever owns it isn’t willing to sell it to you.

You are asking the wrong question.

Finding uninhabiteds island is not a problem, there are countless ones in Oceania or the Caribbean.

Marooning was the term of leaving someone on an uninhabited island and it was considered a capital punishment.

The reason is simple: What do you want to do there what you cannot do with a big yacht or a house on a inhabitated island ? You have already the precondition that you have enough money. With a big yacht you can travel everywhere, replenish stocks and do what you please. A house on an inhabited island gives you social life to enjoy.

Living on an island grows stale very quickly. Most uninhabited islands are for a reason uninhabited: They cannot sustain life for one or several persons, so replenishment of stocks and therefore contact with the outer world are a necessity. If you cannot avoid contact why living on an island (instead on a yacht or house of an inhabited island) in the first place ?

What about friends ? They may find it cool to party some time, but sooner or later a
problem occurs: Cabin fever. Every time a group is isolated and have no possibility to withdraw, it causes irritation and finally aggression.

Anyway, all islands are under a jurisdiction. While officials may never find out what you are doing, once they know they can forcibly evict and punish you. And the law also applies for everything you have done, so no room for misbehavior. User cpast provided in the comments an example:The attempt to elevate the unoccupied Minerva Reefs near Tonga to an own nation. Tonga forced the claimants to abandon their attempt.

Forget the idea.

I suppose it depends whose the island belongs to. If you find a completely unknown island, I suppose it will be yours, or at least you can claim it. If it belongs to another country, you will have to respect the law of this country, in terms of settlement and building construction.

If you take the example of Clipperton island, a French territory with no inhabitant, the French law applies and anyone who wants to access it or start a business (most likely to use its natural resources) should apply to the French authority:

[le] Haut-commissaire de la République en Polynésie française, représentant de l’État, à qui il appartenait d’accorder des autorisations aux particuliers désirant aborder l’atoll ou y obtenir des concessions d’exploitation.

The Wikipedia article still mentions that many smugglers and fishermen most likely come to the island and a French military boat sometimes comes and, among other duties, replaces the vandalised French flag and commemorative stone.