"The film Fargo is not actually "Based on a true story". The Coens later admitted that they added that disclaimer so the viewer would be more willing to suspend disbelief in the story. (An urban legend even says that people have gone to search Minnesota for the briefcase of money.)
While the specific crimes in the movie didn't happen, the plot has elements of two well-known Minnesota crimes. In 1962, a St. Paul attorney named Eugene Thompson hired someone to kill his wife, Carol. Unbeknownst to Thompson, his man hired someone else to do the job. The second man fatally wounded Mrs. Thomspon in her house, but she managed to escape him. She went to a neighbor's house for help while her assailant fled the scene. The sloppiness and brutality of the crime attracted great attention. The murderers were quickly caught and gave up Thompson, who denied knowing anything about the crime for many years afterward.
In 1972, Virginia Piper, the wife of a wealthy Orono banker, was kidnapped. A million-dollar ransom was paid, one of the largest in U.S. history. Mrs. Piper was found tied to a tree in a state park. Two men were convicted of the crime, but were acquitted after a re-trial. One of them later went on a shooting spree after his wife left him, killing her, their 5-year-old son, her son from a previous marriage, her new boyfriend, and one of his sons. Only $4,000 of the money was ever recovered.