The story goes that if you do, it will stay in your digestive track for seven years. Is this true?

According to the Mayo Clinic,

Although chewing gum is designed to be chewed and not swallowed, it generally isn’t harmful if swallowed. Folklore suggests that swallowed gum sits in your stomach for seven years before it can be digested. But this isn’t true. If you swallow gum, it’s true that your body can’t digest it. But the gum doesn’t stay in your stomach. It moves relatively intact through your digestive system and is excreted in your stool.

On rare occasions, large amounts of swallowed gum combined with constipation have blocked intestines in children. It’s for this reason that frequent swallowing of chewing gum should be discouraged, especially in children.

The American Chemical Society has a nice video explaining what happens when you swallow gum: