While I can only speak for myself, it is VERY possible to have rare, expensive games in a storage unit. I have about 90% of my collection in a unit in Virginia (recently moved to San Diego) and could not bring them with me right away. While it is HIGHLY unlikely I would stop paying for my unit and have thousands of dollars worth of games be lost, if I were to get in an accident, die, go bankrupt, etc I could see that being a struggle. And I have some expensive titles in there...Earthbound, Zombie Nation, Snatcher, to name a few. So while uncommon, I can see this sort of thing happening.
As for the episode itself? I saw it. It's likely fake. One thing that always gets me is when they find some rare item, and nothing else in the unit even compares to it in terms of style or value.