This is a very basic list without much info but it should get you started on places you could stay!
I must also mention that since the University is very centralised in Galway, the max you would have to walk from the majority of these places is 20mins. To use a reference, it takes 15-20 minutes to get from the bus station in Galway to the university.
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Bothar na mBan
Kinlay House Hostel
Galway City Hostel
Barnacles Hostel Galway
091 568 644