Where can you find a free house on oblivion

To put it bluntly, there aren't any. You can use any miscellaneus cave or hut or bandit camp you find in the wilderness as a base of operations. Be warned that items you put in nearby chests or barrels etc may disappear. Also, if you decide to live in a cave or camp, bandits or monsters you killed to obtain it WILL respawn sooner or later. And it is a big shock to fast travel to your trusty cave home to find a minotaur staring you in the face.

For something a bit safer however, there is the Roxy inn, north of the imperial city where you can pick up a quest that concerns the village Aleswell. I won't spoil it, but if you complete the quest and don't make the villagers hate you, then you will get a room at the Aleswell inn forever, free of charge. You can return as many times as you like for no charge. However, it is still an inn, so if you try keeping items in your room, they may disappear.

If you join the dark brotherhood, you can sleep at the sanctuary rooms (to the left from the entrance hall) free of charge. The same goes for the fighter's guild and the mages guild. The thieves guild does not have a fixed place you can sleep until you finish the questline.
Or, if none of these options appeal, there is a house on the imperial waterfront for sale (you can buy it at the imperial commerce and trade building in the market district) which costs only 2000 gold. The furnishings are extra however though to start with it comes with a bed and a chest of drawers.