It was the story of a young man named Paul who was enamored with a French pop singer. She is more interested in her career and he either has an accident or commits suicide. According to a December, 1965 interview with Chantal Goya, the actress who played the pop singer, the Paul character commits suicide.
Take a look at the frame from the scene at the police station. where Madeleine (the pop singer) and her friend Catherine are giving statements about Paul's death.
In the upper left hand corner of the frame, there is a poster and the frame cuts off the head of the man in the poster. Is the framing of the scene a coincidence? No. According to an April, 2005 interview with Willy Kurant director of photography for the film:
"Godard asked for alot of odd framings, explaining to me he was wanting the additional person in the frame without being in the frame. He didn't mind or he was wanting them 'chopped in two', just with having an arm or a nose. Alot of directors have absolutely no position on the framing. They don't know what they want. When they arrive on the set, they tell, 'What are we doing today?' That's not what Godard was saying. Godard had a point of view."
Now look at the photo of Paul McCartney in the Beatles' famous 1963 collarless jackets. A resemblance to the poster?
The French words on the poster -- "avec nous" means "with us" so the whole statement on the poster is: "with us, EZ".
I think this film was about the REAL Paul McCartney and his real or virtual (in the old-fashioned sense of the word) suicide sometime in late 1965 or the early 1966.