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First things first: a Greek law dating from 1976 forbids wild camping. Quoting from an article on The Guardian:
Greece
Wild camping here is illegal, the authorities preferring to point tourists towards money-making, taxable hotels and campsites. This rule is rampantly ignored, but we're not going to recommend that you wild camp, just in case. But a friend of a friend of a friend tells us there are numerous stunning spots on the Cyclades islands which are virtually un-policed, particularly the north-western coast of Amorgos and the southern beaches of Anafi. And we certainly wouldn't recommend the east coast of the Peloponnese around Leonidi and Kyparissi, despite the fact that they are totally unregulated and utterly remote. Nope, not us.
The law prohibiting wild camping has a number of caveats including the distinction between motorhomes and campervans, the possibility of sleeping in an open tent which might or might not be considered as wild camping, and whether or not a taking a nap in a sleeping bag is considered camping.
Having said this, let's turn our attention to the logistic aspects of this. The only way to access Navagio beach is by boat. You would therefore have to pay for a boat ride, and convince the skipper to leave you on the island. Another option would be to hide away on the beach hoping that nobody notices your absence. How about the return journey? You'd have to convince a different skipper to pick you up (for a fee, of course). Or you could have arranged something with your departure boatman. All in all this plan depends on the cooperation of the skipper who takes you to the beach. I couldn't find anything regarding rules and regulation that might prohibit boating companies from leaving passengers on the beach overnight. Might be worth a try. Another option would be to get there with your own boat, moor and camp on the beach.
At this point you'll have to hope that the coast guard doesn't patrol the beach. You might get lucky and spend a wonderful night, or you might end up having to discuss with the Greek law enforcement officers - an experience I don't wish to anyone.
If you do end up camping there please be respectful of the environment, and leave no trace of your passage.