I think I had to pay between 70 and 80 euros for a tiny single room with a view of an air shaft just off Placa Catalunya in 2016. (It might have been higher than that; I'm traveling and don't have access to my records.) I think it's very unlikely you'll find a double with a view in your price range that's located near your target area--unless it has really major problems. I just took a quick look at Booking.com, and rates are higher than I was expecting. I wonder whether there's something special happening during the period of your visit.
Be careful about options like Airbnb. Barcelona doesn't seem to be as far along as Paris, but it is reportedly going after illegal (unregistered) rentals. I haven't kept up with the details, but the situation has been discussed on this forum.
Perhaps you can find something farther out near a Metro station.