I was wonder if you could use the chests in the abandoned shack you go to for the Dark Brotherhood. No one's ever out there, but I'm wondering if maybe people don't actually have to be there, and things will just go missing randomly. Is it a safe place to put my stuff?

im going to say no, since the only safe storage places are usually the ones you own. i wouldnt risk it, anyway.

Step 1: Buy a dagger
Step 2: Put said dagger in chest in shack.
Step 3: Go out adventuring for a couple in game (or out of game) days. OR just sleep... a lot... like... weeks
Step 4: Go check to see if dagger is still there
Step 5: ?????
Step 6: Profit! (or feel sad you lost a dagger, either or)

Good question, i'm currently using the house of horrors in markarth for the same reason. The local merchants had pretty much ran out of money so i stashed a few goodies in there including some dragon bones that were taking up far too much weight. Not been back in a while to see if they're still there.

My character isn't the kind of guy who wants to buy a house anywhere so would love a safe place to stash my loot.

As stated by the Devs, the only Safe containers are the ones you own. That means the houses you buy. It also means any homes that are owned by Friends of yours. You know this because the NPC says "My home is yours to use" and nothing in the house is marked as owned. Also, the factions will give you a room or bed to use that has containers in it.

However, I would not put anything in a chest that is not in a home I purchased.

If you're on PC you can set the ownership of the containers.
Alternatively, you can simpy check the UESP if they are safe.