The story begins when Hammy sees his television suddenly go off — a beaver named Boris has gnawed through a telegraph pole, cutting off the signal. Determined to restore his favorite entertainment, Hammy embarks on a mission that leads him through mini golf courses filled with obstacles, objects to destroy, and mechanisms to activate. Although he doesn't hit the ball himself, he runs around the courses to move objects and clear paths for the other characters.

The game features traditional golf holes and special courses where the goal is to break objects with the ball. There is a Challenge Mode with more difficult courses, as well as multiplayer for up to 4 players with turn-based control on the same console. With a humorous approach and simple gameplay, Hammy Goes Nuts! offers a lighthearted experience based on the world of the movie.