If Gilligan's Island were real, what would its size be in square miles?

[TMC]

Given the fact that the island had:

1. A lagoon
2. Caves
3. A volcano
4. Fresh water in abundance enough to support seven people.
5. Food in abundance enough to support seven people.
6. A range of animals from birds to mammals

Is it safe to presume that it was at least 20–30 square miles in area? If so, how or why an island of that size which can support human life was not already inhabited? Especially considering how close that it was to the Hawaiian island chain and how much smaller bodies of land are inhabited in more remote areas (Pitcairn Island, for example).