dmc123,
During my last vacation to Madrid, I stayed at an apartment run by Hostal Madrid,
www.hostal-madrid.info. It was very centrally located. I wrote a review years ago.
I believe it cost almost twice what a room at the hostal would have, but for all the reasons Steve points out, I really enjoyed it. Plus it had facilities for laundry which is always a problem when traveling in Europe.
Gazpacho