Is it possible to buy a house in oblivion 4

Yes, in any of the major cities, go into the castle and talk to the king/queen that is there. You have to be in pretty good standing with them, or else they won't offer it, but you can just keep asking them, so don't be afraid to try. Also, the houses are pretty expensive, the cheapest one you can get is in the city of Anvil, for 5,000 gold. But be warned, this house opens up a quest that you must beat before you can do anything more. It's not a terribly difficult quest, but there is a boss you must beat, and many ghosts/undead.