There is an LGBT society, one overtly gay-friendly club and one overtly gay-friendly pub. (I say overtly because you don't necessarily need a gay bar or a pub with rainbows in its windows to be gay-friendly.)
Because of the size of the city, its geographical location, and the fact that Exeter is in many ways an overgrown rural market town, you will find that the gay scene is nothing like a vibrant as that which you might find in Manchester/Brighton/parts of London. In particular there are no enclaves where you are safe to be as gay as you want without worrying about the presence of prejudiced individuals.
My guess is that you would probably want to be more covert about your sexuality when out an about... but for the real scoop you'd need to ask an actual gay person.
To put it in less PC terms... if you dress as Dafydd Dafydd and march up Fore Street pinching your boyfriends butt, you cannot be assured of the same joyful acceptance that you might be in other parts of the country.
But since hardly any gay men are like that I doubt it'll be an issue. (And of course if you're an ostensibly straight rugby player then that sort of behaviour is just fine... go figure)