The tourist office at Oslo S Station can book you in anything from a luxury hotel to a room in a private home for a fee of NKr 45. Usually there are last-minute discount rooms.
If you want to rent an apartment, contact Oslo Apartments, open weekdays 8-4. Most of their 90 apartments are reasonably close to downtown Oslo, but there are also a few in Majorstuen, 5 minutes away. There are also some in Skyen, a grassy suburban area that's closer to both the airport and city center. All are within a 10-minute walk from public transport.
Oslo Apartments (St. Edmundsv. 37, Skyen. 22-51-02-50. 22-51-02-59).