Fellow gardeners, Don and Lane, participated in growing out pomodorini this past season. These recent photographs show their Principe Borghese tomatoes in storage. They uprooted the entire plant, hung it upside down in their pantry, and are picking the pomodorini off as they ripen. They’ve saved seeds from one of the ripest specimens for another go round next season.
Diary of a Tomato 
Thats great – I’m going to try this this year, although I suspect my partner may not be pleased with my using up too much shed space……
Per Liz’s comment- can they be stored in a shed? We’re in the process of turning our small, detached garage into an outdoor dining room, and I’ve been interested in growing and storing these types of tomatoes for some time now.
I’ve been told that in Napoli they’re stored traditionally in the attic, so it’ll depend on how cold your shed gets. We’ve been storing them in the basement where it’s around 60°F with some success, though I wish we could get the temp further down. It’s been trial and error, but worth a try — the only way to know if it’ll work for you!