As my girlfriend had studied at the University of Tartu, we were keen to stay in one of the student dorms when we were there. In the event, I was able to snag the suite in the Pepleri dorm, which was fantastic value.
The suite consists of a living room, with a fridge, microwave, hot plate, TV and stereo, a separate bedroom, a sauna and a bathroom. Breakfast isn't included, but with cooking facilities it's quite possible to self-cater. We took the even easier route of buying what we wanted each day at the supermarket.
As hostels, or even student dorms, go, this was a remarkably well-appointed place, and the level of security was great as well. Even though the front desk wasn't always attended, it would have taken a lot of effort for anyone to get into any rooms without being meant to do so.
The major drawback, though, was the location. Pepleri is handy to the train station (a great benefit when we arrived in pouring rain), but a fair distance from the main sights of Tartu, and also from the bus station from which we left.. Still, Tartu isn't a big city by any means once you're in the centre, and needing to walk a bit every day isn't a major issue.