LockerBarcelona is less than a five minute walk from the main transport hub of Plaça de Catalunya, and has lockers in various sizes for baggage or shopping or anything you don't feel like carrying around, with code operated locks in English and security cameras.
They offer internet service, will print out your boarding pass, weigh your luggage and have a changing room. Prices start at €4.50 a day.
Also see Lock and Be Free or Locker in the City for other options.
Image by Locker Barcelona