We highly recommend Francois place. We stayed here for 4 nights. It was spotlessly clean, fully equipped ( except for an iron, which their insurance doesn't allow). It very central but at the same time the apartment is very quiet. It's 50 meters from Catalina Square and round the corner from the Gothic Quarter. There are homeless people sleeping on the street very close by ( as there is in most major cities), they did not bother us at all.
There is a fabulous supermarket ( including a gourmet section) under the department store on the corner, selling everything you may need including the best wines and deli counters.
They bed was extremely comfortable and the apartment was absolutely silent.
Its very safe with 3 locked doors before you are in the apartment.
The apartment is up 2 flights of stairs but there is a lift.
It was great to have a washing machine and a balcony to hang the washing out.
The shower was great and lots of hot water and the aircon worked well.
Francois was extremely helpful and responded to our questions really quickly.
We will definitely stay again if we are back in Barcelona.