>>2805391A week is definitely enough.
The city itself is geographically small, so it honestly doesn’t matter where you stay. Best way to get around is via bike, you can cross the city in like 40 minutes fairly easily. Plus the grid pattern of the place makes getting around super easy no matter where you are.
In terms of my personal recommendations of where to stay, it depends on what you’re after. Every neighbourhood IMO has its pros and cons.
You’ve got super central spots like Barri Gòtic. Can be beautiful, also can be sketchy at night, lots of bars and very noisy at night.
If you want the same, just less sketchy, stay in Born. If you don’t mind sketchy and want central but affordable, stay in Raval.
If you want elegance, space, but less excitement after say 12pm (unless you’re gay and also like gay cocktail bars) stay in Eixample.
My true recommendation if I had to give one though would be either San Antoni or Poblenou. Close enough to everything but also very nice neighbourhoods without too much chaos in either.