The film depicts a future in which human beings, known as "Oms" (a homonym of the French-language word hommes, meaning men), are creatures on the Draags' home planet. The Draags are an alien species which is humanoid in shape but a hundred times larger than humans and they live much longer than human beings. Although some Oms are domesticated as pets, they are seen as pests and are periodically exterminated.

A group of Draag children accidentally kill an Om woman during play. Unfortunately her death leaves an orphaned infant, who is taken in by an adult Draag as a pet for his child, Tiva. Tiva’s father just happens to be master Sinh, the Draag great Aedile and after some time, when the child and pet are playing, they surprise him and several of his compatriots during a ritual melding session. It is revealed that many Draag children have Oms like Tiva's.

The bond created between the Draag child, Tiva, and the Om, named Terr (word play on the French word Terre, meaning Earth) deepens as time passes by. Tiva's education is supplied by the use of a headset that transmits knowledge directly into the brain of the user. Because she enjoys having Terr in her hand when she is having her "infos," Terr begins to acquire the Draag knowledge. Terr begins to realize who and what he is, and escapes, taking the headset with him.

He eventually finds other Oms and after some tribulation, is accepted into a tribe. Over the next several scenes, it is shown how the Oms have adapted to life on the Draags' planet. One day, the now-literate Oms reads a new sign on one of the walls, and learns the park is about to be "de-Omized." The de-Omizing is accomplished using disks that release a poison gas. A great many Oms perish from this gas, but a sizable number still manage to escape.

The Oms retaliate and manage to kill one of their Draag attackers. The death of the Draag puts the Council in an uproar. De-Omizing is stepped up to a much higher priority, new technologies are developed, and extermination frequency greatly increases.

Fatalities resulting from Draag attempts to de-Omize are minimized by the creation and organized use of shelters, but the Draags' updated technologies become ever more aggressive, and when an automated scout detects the persistent Om settlement, it summons an array of lethal devices. The Oms launch manned rockets toward the Fantastic Planet, where they discover headless humanoid statues. As Draag meditation bubbles descend to alight atop the statues, the statues begin to dance. This is the secret that animates the statues and allows the Draags to reproduce. When the feet of the dancing statues threaten the rockets, the Oms use disintegration weapons to shatter the statues, which in turn makes thousands of meditative Draag go insane. Pandemonium reigns in the Council chamber, for it seems the two species will destroy one another if they cannot find a way to live together. While the Draag council continues to think of revenge, it is proposed that the two species finally create peace between each other.

The last scene proves that peace has been made as an Om steps down off an outstretched Draag hand, removes his silly hat and assumes a posture of confidence and self-assertion.