In a futuristic city, a taxi cruiser spirals its way to the top of a very, very tall building. It lands on the 17,000th floor. Its oviparous occupant steps out and enters the office of Dr. I.Q. Hi, Secretary of the Stratosphere.
The duck meets Dr. Hi and the two immediately begin ascending on a small platform. Dr. Hi explains that the reason he has summoned his guest is that the world's supply of Alludium Phosdex (the shaving cream atom) is "alarmingly low." It seems that the only remaining source is on Planet X, in a region marked "unknown" as Dr. Hi points out on the rather large wall map.
The duck asserts his confidence in retrieving the material, because "There's no one knows his way around outer space like.... DUCK DODGERS IN THE 24-1/2TH CENTURY!" Dodgers accidentally falls from the platform, plunging several hundred feet, but miraculously survives the accident.
Dodgers uses an evaporator to get to the airport's launch platform, where his ship and his first mate, a pig that Dodgers describes as an "eager young space cadet," await. Dodgers launches the ship, after one false start in which he accidentally plunged his vessel into the ground.
On board, Dodgers explains the navigational techniques he will use to find the planet to his number one, who appears to have a better understanding of navigational techniques than his superior officer, though his stuttering demeanor gets in the way of simple communication. The pig then suggests an alternate method of navigation to Planet X (by following the trail of Planet A, Planet B, Planet C, and so on) to Dodgers, who dismisses the idea at first, but then claims the idea for his own, much to the chagrin of the space cadet.
After weaving its way past Planets R, S, T, U, V, and W, the ship finally lands on Planet X. The ship is marked by a large "X" shape across its surface (presumably a continent) but also bears the "X" mark in several monoliths, presumably the markings of a long-dead civilization of millenia past. Immediately upon landing, Dodgers pulls out a flag of the Earth Alliance and proclaims "I claim this planet in the name of the Earth!"
Unfortunately, Dodgers's plans are immediately thwarted when another ship, christened the Martian Maggot, lands nearby. Its occupant scrambles out and, within meters of Dodgers, claims the planet in the name of Mars.
Needless to say, a conflict develops between the two representatives of their governments. Dodgers claims that the planet simply isn't "big enough for the two of us." The Martian responds by pointing a disintegrator pistol at Dodgers, who believes he is immune because he is wearing a disintegrator-proof vest. Answering a dare, the Martian fires upon Dodgers. Ironically, the vest itself turns out to be disintegrator-proof; Dodgers, unfortunately, falls into a crumble on the ground. The vest follows shortly thereafter.
Fortunately, the space cadet -- and technology -- save the day, thanks to an Acme Integrating pistol. Dodgers, fully reconstituted, continues his territorial challenge upon the alien and threatens him with his own brand of disintegrating pistol. Unfortunately, the pistol itself disintegrates when he attempts to fire upon his opponent. The Martian fires upon Dodgers several times and misses, subsequently chasing Dodgers into his own ship.
In an attempt to deceive the martian, the first officer creates a ruse: pretend to give the Martian a birthday present, assuming that the extraterrestrial will be ignorant of explosives. The Martian gladly accepts the gift, a stick of dynamite with a very short fuse, and is caught in the explosion. Only his helmet saves him from certain death. The alien, visibly shortened by the effects of the blast, limps back to his own ship to attend to his injuries.
Meanwhile, Dodgers is furious at the Martian's resolve. He sends an ultimatum, using an ultimatum dispatcher which behaves much like a gun, but delivers a written message: "Surrender, or be blown into 17,670,002 micro-cells." The martian responds with an ultimatum answerer, which functions much like the dispatcher, only it releases a destructive charge, presumably refusing the terms of the ultimatum.
Not ready to give up, Dodgers decides to conduct espionage upon his counterpart by using the "Super Video Detecto Set" installed in his ship. The advanced state of technology he uses backfires on him, however, because it enables the Martian to fire upon Dodgers via the video screen. This is the last straw for the protaganist.
In a final act of desperation, Dodgers decides to completely annihilate his enemy by utilizing his secret weapon, manufactured by the Acme Destructo Company. He launches a line which casts a dangerous net over his opponent's ship. Unknown beknownst to either of the Earth ships occupants, the alien launches a similar weapon of his own immediately thereafter. The tension mounts as both aliens prepare to destroy their enemy. A tremendous blast is seen and heard; evidently, both systems were set off at the exact moment.
The effect is nothing short of devastating. The smoke clears to reveal Dodgers and the Martian, face to face, standing on the only remaining chunk of Planet X, measuring less than a meter across. Dodgers reiterates his claim that the planet is not big enough for the two of them, and unceremoniously pushes the alien over the edge.
As the camera's aperature narrows, it pans down to reveal Dodger's number one, barely hanging on to some root-like material still attached to the miniscule remains of the planet. The martian, precariously hanging onto the first mate, looks on as the pig proclaims, with a slight stutter, "Big Deal."