Where can I safely store my stuff?

I'd like to find a place to store stuff while I'm out adventuring, without spending 5000 on a house (at least, until I can afford that...). Where can I safely store items without worrying about them disappearing?

There are two possible places that come to mind, Gerdur's house (to which I have the key and was told I could stay), and the small bedroom given to me at the College of Winterhold (which I was told was mine, despite seeing J'zaro sitting in it later). Can I store items safely in either of these places?

Places you can safely store your items include any houses you buy, your companion(s), and your horse.

Places considered to be owned by you (and therefore contain safe containers) are (spoiler warning):

• Arch-Mage Quarters (after you become Arch-Mage)
• The Dawnstar Dark Brotherhood Sanctuary (After you've finished the Dark Brotherhood quest line)

I have personally stored items in my apprentice quarters in the College of Winterhold without problems, however others have had stuff go missing from there, so be careful.

Other containers may not be safe. Given what some other answers have said, Gerder's house does not seem like a safe place for long-term storage.

The Skyrim manual states:

WARNING! Do not leave things in containers you do not own! They may not be there when you come back for them. Buying a house is one way of owning containers.

I, and Doug, have also stored items in the room assigned to you at Winterhold College and it appears safe.

EDIT: As stated in the comments below, the Winterhold College room is not safe. It just seemed safe to me at the time when the game was very new.

Your best bet is to get a house. If you need additional space now, maybe try storing the items in your horse or companion, or try out those two locations you mentioned with something inexpensive. See if it disappears.