If you really want to be away from the tourists and yet have easy access to Placa Catalunya, Barri Gotico and the Born, it's Sarria or Pedralbes with Gracia right in there as well. If you go to various hotel websites, look for "Zona Alta." That's the "high zone," literally the neighborhood higher up toward the mountains behind the city. Sarria is now a pedestrian-only historic district, that began as the location of summer farms and villas, so that one could escape the summer dampness and be in "higher" (read fresher) air in the summertime. Many of the old summer "masias" (villas) still exist, and there are a few hotels on its outskirts; Pedralbes is a leafy outskirting suburb--yet with a metro stop nearby, easily accessible to downtown. It's where the Barca futbol/soccer players and their families all live because it is close to Camp Nou, the Barca stadium. Also there's the incredible L'Illa shopping center on Diagnal and the most incredible Rose Garden higher up on Diagonal before one gets to the Rey Juan Carlos Hotel and polo grounds. The Garden started with roses found in Spain and now has expanded to include roses from all of Europe and Asia, etc. Their long-term goal is to include roses from all over the world.
Hope this helps.