I'm looking for a beginner house to store unique items and other stuff, at level 1.
Anise's Cabin fits the bill. The house is Southwest of Riverwood, across White River, South of Bleak Falls Barrow.
The house contains a bed, an arcane enchanter and alchemy lab as well as several containers.
The Cabin is owned by Anise, but if you go into the cellar inside the cabin, and then exit, she will attack you. You can then kill her and claim the cabin. The bed becomes un-owned, but everything else is still owned, so either fence the items, or drop them somewhere.
EDIT: Jim reminded me about the containers. The containers in the basement do not reset. Most of the ones above ground are also safe, but the barrel is not. To be safe, only store stuff in the basement.