Sadly, bars - much like restaurants - are in short to zero supply around the circuit. What you can do is travel back towards wherever you're staying (probably Cheste or Valencia) and find a huge number of bars there. Valencia has dozens, of course, and options to suit every need, whether it's cocktails, pints, outdoors, black tie - you name it! Cheste is much the same but on a smaller scale - lovely, tasteful bars with experienced and friendly staff. Go look around!

Since the circuit is in the middle of nowhere, it's not going to be easy to find bars around the venue. The best option in my opinion is to go to Cheste (nearest town, 9 km) and then try to find a place to have a drink. In Avenida Castilla and surroundings, go to a bar called Castilla, nice for tapas and for drinks. There is also a place for kebab called Bar Stefani which is very popular in the town.