If you are looking for a real quiet place, you have to visit Telendos. This small island is situated close to Kalymnos.
NIn Mastichari you take the ferry to Kalymnos (9 euro a person for a return ticket). At the harbour over there take a taxi and tell the driver you want to go to Telendos. For 12 euro he brings you to Myrtees, a small harbour were every 30 minutes a fisherboat brings you to Telendos within 10 minutes for 2 euro a person.
I haven’t seen even 1 car, moped or bicycle at Telendos. Along the harbour and the restaurants there is a kind of boulevard. The rest of the island only has sandy roads.
I talked to people over there who spent yearly 3-4 weeks on Telendos. Personally I think it is too quiet. Nevertheless the island is worth visiting to enjoy the nice, quiet places before you see the hectic life again when you return at the harbour in Kalymnos.