Barcelona is large(ish) but "compact". Nothing is "too far" from everything else, be on foot, be via public transport (yeah, we use these here, and they're perfectly all right, safe, clean, etc. LOL!)
Another characteristic of most European capitals is that there aren't many "residential-only" areas per se within the city limits if any at all. All neighbourhoods are a mix of commercial+residential therefore all of them have shops, restaurants, etc.
In fact, in the case of some cities, like Barcelona, many of today's neighbourhoods were existing towns that were "absorbed" as the city grew. In many cases, these towns dated several hundred years back, thus, they already had a web of shops, restaurants and other services.
Equally, sights and museums and other places frequented by tourists tend to be spread across several neighbourhoods of the city, which demands to move from one to another if you want to see them.
Lastly, also a tip from visitors from some US states: prepare to walk from A to B, and occasionally to take the metro, bus or tramway. Forget cars while in the city: traffic is horrible, parking is expensive and difficult to find, some neighbourhoods are semi-pedestrianised, etc...
Tip: in Barcelona, as in many other large cities in Europe (ie Paris, Brussels...), the lower the zip code, the more central. Thus for Barcelona, 08001 is "the heart" of the Old City.