We walked from the Sagrada Familia to the beach via L'Eixample district and it was a lovely neighbourhood with good bars and restaurants. We had quite a leisurely walk and this neighbourhood was interesting and unthreatening, unlike some areas we have visited. I suppose like all cities, though, you need to be careful at night, especially if you are a stranger.
The area looked clean and we would have had no qualms with staying in any accommodation in the area. We may even stay here if we return.