Of everywhere I traveled, my favorite place in the world to visit was Samosir Island in Lake Toba.
So it's a lake island in a volcanic lake on the Island of Sumatra in the worlds biggest Island country (Indonesia).
Its not a private Island and there are people living there and even tourists coming regularly. But getting to it is quite the effort, so I call it reclusive in that way.
It's unfortunelately really hard to get a building permit on a island. They're virtually always vacation home plots. Funnily enough though islands are usually cheaper than the equivalent size of land on the mainland.