I came to Tartu for a conference and I stayed at the Pepleri Hostel. I don't know if I was lucky or if all the rooms are like this, but I got a double room with a small kitchenette, a small bathroom and a TV, all for 25€ per night (breakfast not included). Not an elegant place by any means, but very convenient specially for that price.
The hostel is not exactly in Tartu city centre, but Tartu is not that big so that is not a big problem.
What I cannot imagine is living in this hostel permanently, as some people apparently do, specially if you share a room... But for a short stay, and if you are not looking for something fancy, it's very good.