I think the other guys said everything.
250 euros isn't enough to rent an apartment, you'll need at least 300 euros for an old one.
There are no hostels in Thessaloniki, only hotels. But I don't know any because I've never stayed at one :) :)
You should also know that ATEI are located several km away from the center of Thessaloniki. It is more than 30 minutes trip by a bus.
Hope I helped you.