It's a real museum in Croatia (there's an installation in L.A. too!) which, says Wiki "began as a traveling collection of donated items." I came across it via an article in VQR.
The Museum of Broken Relationships is a collection of ordinary objects hung on walls, tucked under glass, backlit on pedestals: a toaster, a child’s pedal car, a modem handmade in 1988. A wooden toilet paper dispenser. A positive pregnancy stick. A positive drug test. An axe. They come from Taipei, from Slovenia, from Colorado, from Manila, all donated by strangers, each accompanied by a story: In the 14 days of her holiday, every day I axed one piece of her furniture.
One of the most popular items in the gift shop is the “Bad Memories Eraser”—an actual eraser sold in several shades—but in truth the museum is something closer to the psychic opposite of an eraser. Every one of its objects insists that something was, rather than trying to make it disappear. (Source)
Here's a TEDx talk from the museum's creator, Olinka Vištica.
Vištica's book -- photos of donated objects and the accompanying breakup stories -- came out last year. Amazon's description says the museum now has a permanent installation in Los Angeles.