“Both” refers to two items only. It is easy in speech to absent-mindedly add items to an initial pair and wind up saying things like “I like both mangos and papayas and Asian pears.” Try to avoid this when writing.
How do you use “both” in a possessive construction? It’s not easy. “It’s both of our home town” sounds awkward. Better to restructure the sentence and say “it’s the home town of both of us.”
People occasionally say things like “I phoned both them,” when they mean “I phoned both of them,” or “I phoned them both.”