Secret agent Lemmy Caution (Eddie Constantine) enters Alphaville in his Ford Galaxie. He has crossed intergalactic space to visit the city/planet. He checks into a hotel as Ivan Johnson, a reporter for the Figaro-Pravda. He's shown to his room by a Seductress Third Class who starts to undress and offers to join him in the bath. He's attacked by a thug who's entered his room, but Lemmy manages to defend himself. The seductress reclines in the bath, unsurprised by this - it happens all the time. Lemmy tracks down a previous agent also sent by the Outer Colonies, then tries to find Professor von Braun, the creator of Alpha-60 (the sentient computer that runs Alphaville by pure logic), and finally tries to disable Alpha-60.
If that didn't make any sense, trust me, watching the movie itself makes even less sense. What frustrates me most is that people don't act like people: they act like philosophy- or poetry-spewing automaton moving about through what was - in 1965 - modernist architecture and talking about the virtues of logic over emotion (or vice versa). Which also makes it very hard to invest in anyone, or to believe that they could fall in love - or to understand why they would fall in love, when no one's actions or words make any sense.
The film has acquired minor classic status in both science fiction and the movie community. I get that it's Jean-Luc Godard applying the French New Wave to a Noir detective film. But finding meaning in it has been deliberately made difficult with the constant quoting of surrealist poetry, and is an emotionally distancing intellectual exercise that doesn't appeal to me in the slightest. I don't mind the time it took to watch it given the influence it's had on movies since, but I wouldn't recommend it to anyone who wasn't in film school.