The uninhabited 4km-long Illa de Sa Dragonera is a ripple of an island that stretches out like a slumbering dragon to the west of Sant Elm. Constituted as a natural park (Parc Natural de Sa Dragonera), the island is accessible by ferry, which lands at a protected natural harbour on the east side of the island. From there you can follow trails to the capes at either end or ascend the Na Pòpia peak (Puig des Far Vell, 352m).
The ferry operators also do glass-bottomed-boat tours around the island. If you want to dive off the island, try Scuba Activa.