It is usually considered that the center of Barcelona is the Plaça Catalunya, from which large commercial areas such as Passeig de Gràcia, Portal de l’Àngel, Pelai or La Rambla start. Too many people and cars.
But right next to Plaça Catalunya there is one of the quietest corners of the city. It is the Church of Santa Anna (Santa Anna 29, but you can also get down the street Rivadeneyra from Plaça Catalunya). They charge to enter, but the cloister is pretty relaxing.
It is worth visiting in the hottest days to rest for a while.