Yes, you can store things in the auction house's Completed tab.
Armor and weapons have a 5-minute "grace period" when listed, so you can list something and immediately cancel it. This will put it on your "Completed" tab, and then you're free to leave it there indefinitely. If you miss this grace period, you'll have to wait to get your item out until the auction times out, and there's a chance that someone will buy it from you and you'll lose it.
When you get to 50 items on the Completed tab, you can no longer list new items for sale. However - things only count towards this 50 when cancelled, and you also can have up to 10 auction at once. Therefore, it's possible to get 59 items into your Completed tab if you have 49 items with no items listed, and then you list + cancel 10 items right away.
Presumably, if you bid on a bunch of items (and eventually won them) while you were close to the cap or before you tried this trick, you could get more items in your Completed tab than this. I don't know that there's much benefit to this, however.
Also note that the only destination for items in this view is your stash - if you're using this to augment the size of your tiny, insignificant stash, you'll have to jump into and out of a game in order to clear room for all the items you've stored should you want to purchase something.
Do note that the Completed tab houses things you've purchased as well as things you've attempted to sell and cancelled or which ran out of time with no winner. Therefore, if you're over the limit, you are unable to bid or buyout on any auctions until you clear some space for new items.
Crafting materials do not seem to count towards the 50 limit - if they end up on this tab (either bought or sold) they won't count. (Do note that there's no grace period for cancellation on commodity items!) Similarly, gold refunded from unsuccessful purchases or for sold items doesn't count towards this limit.
For other related info on the Completed tab, see this question.