To sleep on the beach is complicated because it is closed from 22h to 5h. You also have to pay to access it from 12h to 19h.
A good place to camp can be found at the entrance, near Kadir's tree houses. When you arrive in Olympos, Kadir's tree house is to your left, you walk a few meters, you will have a path to the forest to your right. Follow the path, you will go in the direction of outside of the city. On your right, you will find some places suitable for camping and after a little abandonned house.
You can also go to the next beach in Çıralı, if you go a bit further away from Olympos, you can easily find place to pitch a tent. And if you go a bit more north you can join the Lycian Way, after hiking one and a half hour you will find isolated beaches, really nice to camp, with fresh water from river (at least in winter).
Olympos is a touristic destination on the south coast of Turkey, very close to Antalya. There are some ancient Greek ruins, and a nice beach. You have to pay to access to the ruins and the beach. But it is just next to the village of Çıralı where there is a much bigger beach and from where you can access the beach of Olympos without paying. Some nomads could go inside the ruins from the beach without paying in december 2014.