score:9
Apparently Oxford Backpackers will hold on to them for you, for a fee:
Oxford Backpackers, 9A Hythe Bridge Street, will accept left-luggage even from non-residents at 2 pounds per person per day - but not necessarily per item, if two items can be tied together. Remember, however, that you have to climb a flight of stairs to start with.
This from a link on tripadvisor that I was reading. The hostel is just a few minutes walk from the station, according to the page.
Upvote:5
Just returned from Oxford July 2015. Backpackers Hostel on Hythe Bridge street will hold luggage for the day. They charge £ 4 per item. Is about 1 block from train station. Caution there is a very steep long staircase to get into hostel. Staff was very friendly and welcoming!