The film depicts a future in which human beings, known as "Oms" (a word play on the French-language word hommes, meaning men), have been brought by the giant Draags to the Draags' home planet, where they are kept as pets (with collars). The Draags are an alien race which is humanoid in shape but a hundred times larger than humans, with blue skin, fan-like earlobes and huge, protruding red eyes. The Draags also live much longer than human beings one Draag week equals a human year. Some Oms are domesticated as pets, but others run wild, and are periodically exterminated. The Draags' treatment of the Oms is ironically contrasted with their high level of technological and spiritual development.
The story opens with a woman running, occasionally looking behind her as if pursued. An enormous hand descends and blocks her way. She runs back the way she came and finds her way blocked by another hand. It becomes apparent that she is being toyed with by entities that do not appreciate her fragility, and as she dies, the infant she has been carrying and attempting to protect begins to cry.
The view changes to reveal the Draag children who have accidentally killed the woman; they leave quickly when an adult Draag and child approach. The child voices concern for the orphaned infant, and the two take the child to their home. Tiva (the Draag child) names the infant Terr (word play on "terrible", same spelling in French; also a homophone for the French "Terre", meaning Earth). Her father, whom the adult Terr voice-over explains is master Sinh, the Draag great Aedile, attaches a collar capable of physically dragging Terr back from mischief, and over the next several scenes, their relationships develop.
Terr witnesses the parents seemingly ingesting food by inhaling it from a device. After changing Terr's costume as one would a doll's, Tiva uses makeup to give herself a more Om-like appearance. When Terr impishly trades dark pigment for light, Tiva blows some of the powder on him. Tiva uses a tiny indoor weather-maker to cause a small storm cloud to form over Terr and chase him around the dwelling. Tiva takes Terr for a walk, and then teaches him how, under certain circumstances, crystals will form on stationary objects, including standing bipeds. She also teaches him that whistling will shatter the crystals. Terr happens upon master Sinh as he and several compatriots are melding in a ritual, and it is revealed that many Draag children have Oms like Terr when they convene to watch their respective Oms interact.
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 their knowledge.
Meanwhile, at the seat of government, Draag Councilors discuss whether the regular extermination of the wild Oms is sufficient to keep their numbers at an acceptable level. It is revealed that Oms were first found on a planet that retained some evidence of structured life, but the images seem to reveal that Earth was in a post-apocalyptic state at the time.
Terr decides to escape, and to take the headset with him. He does not get very far before Tiva realizes he is missing, and her mother tells her to use her bracelet to bring him back. Terr finds himself suddenly being dragged backward by the collar. Only the headset becoming entangled in plants allows a wild female Om to come to his rescue before he is choked by the collar or dragged all the way back.
When Terr explains that the headset contains the knowledge of the Draags but he doesn't know where to go with it, his unnamed rescuer takes him to her tribe, who live in a tree in a walled park. When it is demonstrated that Terr can read Draag script, the leader (known only as "Mighty One") is willing to accept Terr into the tribe, but the Wizard is not, and demands a trial by combat to the death. Terr and the Wizard's champion have child-sized animals bound to their torsos in such a way as to prevent the combatants from using anything but the beaks of said animals to attack. Terr is injured, but wins the trial.
Over the next several scenes, it is shown how the Oms have adapted to life on the Draags' planet. Snail-like animals weave clothes onto the Oms, predators that would eat Oms are in turn hunted and efficiently stripped of useful materials, and the gene pool is kept well-mixed. Oms even make the occasional foray into Draag areas in search of resources. Returning from one such expedition, the group of adventurers is accosted by "bandits" who drop clawlike harpoons into the cargo and simply lift it up into their own tree. Mighty One tells Terr that they live on the other side of the park, and cautions him that they are evil.
When the now-literate Oms read the new sign on one of the walls, they realize the park is about to be "de-Omised." Terr decides that he must take this information to the tribe of "bandits," and is quickly captured and taken before their leader, a wizened old woman.
The woman is skeptical of Terr's claims. Terr is tied up and left. But when the de-Omising begins, the old woman returns and frees him. The de-Omising is accomplished using disks that release a poison gas. A great many Oms perish to this gas, but a sizable number still manage to escape through a crack in the park wall.
Two passing Draags witness the exodus, and one begins crushing the Oms underfoot. The Oms retaliate and manage to bring down one of their attackers, but Mighty One is also killed, and the old woman leads the survivors to a place where she believes they will be safe. The death of a Draag puts the Council in an uproar. De-Omising is stepped up to a much higher priority, new technologies are developed, and frequency is scheduled to increase.
The old woman has led the two now-united tribes to an old rocket depot. Applying their newfound knowledge, the Oms, seemingly under Terr's direction, very quickly adapt the abandoned technologies to their own purposes and begin to flourish, thanks to the rebirth of mechanized industry. On a visit to the old woman, Terr's former rescuer hears her express both optimism and regret that she will not live to see the Oms finally find peace.
Fatalities resulting from Draag attempts to de-Om the rocket depot are minimized by the creation and organized use of shelters, but the Draags' updated de-Omising technologies become ever more aggressive, and when an automated scout detects the persistent Om settlement, it summons an array of lethal devices.
As the attacks become more diverse and effective, the Oms launch their manned rockets toward the Fantastic Planet, where they discover headless statues. As bubbles descend to alight atop the statues, the statues begin to dance. Each bubble seems to contain an image of an individual Draag in meditation; their "spirits" are what animate the statues.
It turns out that the statues facilitate "nuptial rites" between the Draags and entities from other galaxies, and from these, the Draags draw their life force. When the feet of the dancing statues threaten the rockets, the Oms use energy weapons that shatter the statues. Pandemonium reigns supreme in the Council chamber, for it seems the two races will destroy one another if they cannot find a way to live together.
But in the very next scene, an Om steps down off an outstretched Draag hand, removes his silly hat and assumes a posture of confidence and self-assertion. The headset voice dispassionately recounts the Oms' construction of a new satellite where they can live, "which they call Terr, after their ancestral planet ."