One person’s analog trash is another person’s vintage treasure. If you have a pile of VHS tapes that haven’t been touched since the dawn of digital media, you might be able to make a fortune on them.

After a long day at the office, Hannah Johnson, a deputy county prosecutor in Indiana, likes to unwind with a movie — so she throws one of the nearly 200 VHS tapes she owns into her VCR player.

It’s been half a century since the VHS (Video Home System) tape was introduced to the world, allowing movie lovers to watch their favorite flicks on demand for the first time. We kept playing (and ...