With a metro pass, you can get almost anywhere in the city pretty quickly. I stayed at the Four Points Diagonal which was a quick 2 minute walk to a tram that took me directly to the beach. 8 minutes to the metro.
Also it seemed like lots of stuff to see everywhere in the city.
But if you really want walking distance - as stated above Gothic Quarter, but you will pay a premium I would suspect. Maybe just a bit further away is La Ribera. Still a good neighborhood and walking distance to the major sites.