Sloppy Joe's Bar is located on Key West's main business street, Duval Street. The bar opened in 1933, the day prohibition ended. It was to move locations and have a name change or two before it ended up being at its current site in Key West. The name finally chosen was borrowed from the original Sloppy Joe's bar in Havana, Cuba, although today the name has really stuck to the Key West location.
Local legend has it that Joe Russell, the original owner, was a friend of author Ernest Hemingway who lived in Key West at that time, and that Joe provided Mr. Hemingway with some illegal hooch during prohibition. It seems likely that Hemingway would have visited the bar when he lived in Key West during the 1930s as it seems to have been very suitable for him—nothing flashy, a place to meet friends, drink lots of (reasonably priced) booze and do a little gambling.
You can order food at Sloppy Joe's, which may even be very good although we ate elsewhere so I can't comment on that. There's live music you can enjoy, but I certainly had the impression that this was a drinking man's bar. No amateurs need apply.
Ever since 1981, the bar has held an annual Ernest Hemingway look-alike contest, which I think would be fun to watch. In 2006, Sloppy Joe's Bar was marked as a historical landmark by the National Register of Historical Places.