Sometime in the 2300s, Earth has run out of illidium phosdex, used to make shaving cream, so the government sends Duck Dodgers (Daffy Duck) and his cadet (Porky Pig) to get some from Planet X (to get there, Duck Dodgers suggests that they do a nonsensical zigzag, but his cadet says that they should just follow Planets A, B, C, etc.). Unfortunately for Duck Dodgers and his cadet (but fortunately for the audience), Marvin the Martian is also trying to claim the planet. After that, the rest of the cartoon consists of Daffy and Marvin trying to destroy each other with progressively wackier and wackier weapons (which was probably an allusion to the arms buildup in the Cold War).
"Duck Dodgers in the 24 1/2th Century" is mostly Daffy and Porky doing their usual crazy stuff, with some great one-liners: "Whoops. Had the silly thing in reverse." and "This planet ain't big enough for the two of us!" This is what classic cartoons are all about!