Tèxtil Cafè is in the El Bourne area, located in front of the Picasso museum. There used to be Textile museum, which moved out. It explains the name of the cafe.
You can count the history of the building back to the 13C, as many of others in that area, built with thick stones.
The sunny terrace, which is in the small patio of the building, is one of my favorite coffee spots in Barcelona.