In a prologue, a Jewish man in an unnamed pre-WWII Eastern European shtetl (possibly in Poland, as suggested by the mention of Lublin and Krak贸w) tells his wife that he was helped on his way home by Reb Groshkover, whom he has invited in for soup. She says Groshkover is dead and must be a dybbuk. Groshkover (Fyvush Finkel) arrives and laughs off the accusation, but she plunges an icepick into his chest. Bleeding, he exits their home into the snowy night.
In 1967, Larry Gopnik (Michael Stuhlbarg) is a professor of physics living in St. Louis Park, Minnesota. His wife, Judith (Sari Lennick), tells him that she needs a get (a Jewish divorce document) so she can marry widower Sy Ableman (Fred Melamed). Meanwhile, their son Danny (Aaron Wolff) owes twenty dollars for marijuana to an intimidating Hebrew school classmate. He has the money, but it is hidden in a transistor radio that was confiscated by his teacher. Daughter Sarah is always washing her hair and going out. Larry's brother, Arthur (Richard Kind), sleeps on the couch and spends his free time filling a notebook with what he calls a "probability map of the universe".
Larry faces an impending vote on his application for tenure, and his department head (Ari Hoptman) lets slip that anonymous letters have urged the committee to deny him. Clive Park, a student worried about losing his scholarship, meets with Larry in his office to argue he should not fail the class. After he leaves, Larry finds an envelope stuffed with cash. When Larry attempts to return it, Clive's father threatens to sue Larry either for defamation if Larry accuses Clive of bribery, or for keeping the money if he does not give him a passing grade.
At the insistence of Judith and Sy, Larry and Arthur move into a nearby motel. Judith empties the couple's bank accounts, leaving Larry penniless, so he enlists the services of a divorce attorney (Adam Arkin). Larry learns Arthur faces charges of solicitation and sodomy.
Larry turns to his Jewish faith for consolation. He consults two rabbis (Simon Helberg and George Wyner) but his synagogue's senior rabbi, Marshak, is never available. Larry and Sy are involved in separate, simultaneous car crashes. Larry is unharmed, but Sy dies. At Judith's insistence, Larry pays for Sy's funeral. At the funeral, Sy is eulogized as "a serious man".
Larry is proud and moved by Danny's bar mitzvah, unaware that his son is under the influence of marijuana. During the service, Judith apologizes to Larry for all the recent trouble and informs him that Sy liked him so much that he even wrote letters to the tenure committee. Danny finally meets with Marshak, in a brief encounter in which Marshak only quotes Jefferson Airplane's "Somebody To Love", names the members of the band and returns the radio while counseling Danny to "be a good boy".
Larry's department head compliments him on Danny's bar mitzvah and hints that he will receive tenure. The mail brings a large bill from Arthur's lawyer. Larry decides to pass Clive, whereupon Larry's doctor calls, asking to see him immediately about the results of a chest X-ray. At the same moment, Danny's teacher struggles to open the emergency shelter as a massive tornado closes in on the school.

Question: Why does Larry's wife need a divorce document?
Answer: She is trying to marry Sy Ableman the widower.
Alternative answer: So she can marry Sy Ableman.