Try searching for "pansiyon" in google. Thats the name for hostel kind of places in Bodrum. ( not dormitory style) they are cheap and very center located.
about 6 years ago
We stayed at Azka hotel which was a short taxi ride by water or land away from downtown. It is on Badaksci Bay, reasonably priced but beds were hard.
almost 6 years ago
Simple and Flexible