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There are plenty of left luggage places in Barcelona city centre - one example is that there are luggage lockers at the EstaciΓ³ del Nord bus terminal, next to Arc de Triomf (https://www.barcelonanord.cat/en/services/passengers/), however if you only have small luggage (hand luggage size, so 55x40x25cm) then most restaurants won't have any problems if you take it with you (I have done this myself a number of times in different cities).
Other alternatives for leaving luggage are left luggage lockers at Barcelona Sants if you're staying in that area and are taking the train to the airport, or Locker Barcelona near Plaça de Catalunya (https://www.barcelona-tourist-guide.com/en/transport/locker-barcelona.html) - I haven't used their service so can't give a personal recommendation though. Google also gives various other options.