Yes, as others have already mentioned, Barcelona isn't a cheap city. It's been a major tourist destination for decades, and places like that rarely come cheap.
That said, there's quite a range of prices, and I'm not entirely sure what you mean by "budget". Everyone has a different idea of what's affordable!
But you've already mentioned something I always recommend: don't insist on staying right in the city centre. A hotel near a metro station can be perfectly convenient, even if it's 20 minutes from the centre. Barcelona has an excellent public transport system. Buses are good too, although they can get caught in traffic, whereas the metro is much more predictable.
Those of us who live here use a simple rule of thumb: allow roughly 2 to 3 minutes between metro stops. So if you're six stops away, you're looking at around 15 to 20 minutes on the train. And trains run every few minutes, so getting around is generally very easy.
Having said that, I wouldn't go to the opposite extreme and book somewhere at the other end of the city just to save a little money. When you're sightseeing all day, especially if it's warm, it's nice to be able to pop back to your hotel, have a rest, and head out again later. That's much less convenient if you're staying too far away.
https://community.ricksteves.com/travel-forum/spain/the-quintessential-question-where-to-stay-in-barcelona
One thing I'd definitely look into is hostals (with an A, not hostels!). They're not necessarily youth hostels or dormitory accommodation. A hostal is more like a very basic hotel, usually without all the extra facilities and services. You get a decent, clean room, somewhere to sleep, and not much else. And they can be a little cheaper than regular hotels. Not dramatically cheaper, mind you, but sometimes enough to make a difference.
Years ago, I actually put together a map of hostals that were frequently recommended by travellers on the Barcelona TripAdvisor forum. I simply collected places where people had stayed and reported good experiences. It's an old map, so some of them may no longer exist, but I imagine quite a few are still around. They're not personal recommendations, just a collection of suggestions from other travellers that might give you somewhere to start looking.
https://www.google.com/maps/d/u/0/edit?mid=1jx2jNqHj2ud1bvwPFAcVQcZ7fME&usp=sharing
One last thing: airbnbs and the like are also an option, but make sure you're booking a properly licensed tourist apartment or an aparthotel. Barcelona has strict regulations on holiday rentals, and it's worth checking that the accommodation is legitimate before booking.