In a part of rural Vermont known as the Northeast Kingdom, the museum sits next to the home of Clare Dolan, its founder, curator and self-described “chief operating philosopher.” An artist and puppeteer, she started the museum in 2011 after clearing out her barn and choosing the first exhibition topic: the match. Some items from that show that remain on display include a violin that a prison inmate made from wooden matchsticks (it really plays), an array of matchbooks from across the globe and a roller coaster reproduction constructed of paper matches.

Her goal is to recognize the extraordinary in the ordinary. “We need to celebrate what is mundane and ordinary but beautiful,” Ms. Dolan said.

Not in a major city or even a town center, the museum occupies a 70-year-old barn in the middle of nowhere. It does not display its collection in a pristine environment on stark walls but in a rickety wooden structure with a leaky roof and no heat.

It has no security guards to protect its art. Indeed, one piece went missing last spring, and a small placard was put in its place, admonishing museumgoers not to steal.