The Metro, or underground, in Barcelona is an excellent and inexpensive way to get around the city. You can also use it to get from the airport to central Barcelona too. TMB that run most of the metro lines, except for three short ones, state that 90% of stations are wheelchair accessible so far with work planned ongoing to others. A larger more usable version of the accessible Metro map above can be found by clicking on this link (updated to February 2019.)
I would suggest printing out a copy of the map to carry with you when travelling, I did and found it invaluable for planning my routes, especially when working out how to get to places if the obvious route included a change of lines that are inaccessible.