The Basque Hotel has several properties that it manages across Donostia. The one I picked--on Calle Salamanca 5--was just ideal. It's a 3 minute walk to the Playa Concha and 2 minutes to old town.
The building is very old and exudes a lot of charm. Thankfully, it has a modern, high speed elevator as the apartment is on the 6th floor.
The only thing you need to be aware is that you must pick up the keys at their offices a few blocks away. What I did was have my taxi driver stop at the office first and wait for me. Since I arrived after hours I needed to enter a code (given to me) to obtain the keys. The actual apartment was less than 5 minutes away by car (20 minutes walking).
Overall it was a great experience and I would not hesitate to book the same apartment again! The whole staff was very nice and responsive over email which was most helpful.