Is it possible to become a barrister without going to Oxbridge?

Yes, which is why University prestige is believed to be correlated with a more rigorous course, and hence a disproportionate number of Pupillages going to candidates from Institutions such as Oxford and Cambridge.

Similarly, the GDL is condoned to allow positive competition and better candidates for Chambers (and Solicitors firms) to choose from, which there perhaps wouldn't otherwise be if it weren't for this route. Maybe, also, the fact that modern Pupillage, and Training Contract, recruitment is tending towards almost 50:50 QLD v. GDL shows that Chambers do recruit pragmatically and understand the merits of reading another Arts, or STEM subject at University.