The film opens with a vivid depiction of Mars and Earth, showing how close the two planets came during the summer of 1954. The narrator, voiced by Daws Butler, explains that this unusual proximity caused a "cosmic force disturbance," leading to a bizarre baby switch: a baby destined for Earth was diverted to Mars, and conversely, a Martian baby was sent to Earth. This story unfolds visually through two comet-like bodies colliding and then diverging along different paths, one heading towards Earth and the other to Mars. The green comet, in particular, travels through Earth's atmosphere, passing over numerous homes with strange-looking TV antennas before finally arriving inside a hospital.
Inside the hospital, Joseph Wilbur, voiced by Daws Butler, anxiously waits in the nursery with other expectant fathers. A voice over the PA system announces that he can now see his baby, fueling his excitement as he presses his face against the nursery window. When the baby is rolled in, Joseph is initially overwhelmed by the sight of a baby with a distinctly green head and two antennae that spark and beep in Morse code. Shocked, he comically exclaims, “Somebody goofed!” and quickly faints, overwhelmed by the strange situation.
The story then shifts to a suburban home where Joseph is arguing with his wife, Martha, voiced by June Foray. He pleads with her to let the baby stay inside, but she insists the infant needs sunlight and fresh air. Reluctantly, Joseph takes the green baby out in a stroller, trying to keep the baby’s unusual nature a secret. While he isn't paying close attention, the baby surprises him again by crawling onto the stroller’s hood and beeping, then scampering onto a nearby wall to communicate with a bee sitting on a flower.
Later, inside the house, Joseph expresses concern to Martha that the baby might talk to any strange insects or animals he encounters outdoors. As he pushes the stroller along the street, an elderly woman, played by [June Foray], begins to dote on the infant, remarking how healthy and green he looks. When she notices the baby’s antennae and hears him beep, she becomes increasingly disturbed, even using a pitch pipe to find the right note, which results in her shrieking loudly. Joseph, noticing her reaction, hurriedly brings the baby back home, visibly tense.
Martha begins to worry as her husband's antics grow more bizarre. The baby is shown doing complex activities such as solving Einstein’s mass-energy equivalence with letter blocks and constructing a fictional molecule called "illudium," famously referenced in Marvin the Martian cartoons. He even creates a Tinkertoy model and builds a planetary system out of household objects, like a basketball and Christmas ornaments. The scene humorously depicts him working on plans for a better mousetrap — blueprints included — while happily playing on the family’s television, watching Captain Schmideo promote a toy flying saucer as part of a cereal promotion.
The plot thickens when the family receives a message from Mars, delivered by a small rocket. Initially believing it’s just a joke or a letter from Martha’s mother, they are stunned when it reveals that the child they are caring for is actually from Mars, named Mot, and that the baby they originally had, called Yob, is now on Mars. The message, from a Martian named Sir U. Tan, humorously references a popular vegetable laxative and explains that the cosmic exchange resulted in the switch, warning them to take good care of the Martian infant until the exchange can be arranged.
In a comical turn, Mot, using his toy spaceship, launches himself out of his highchair like a rocket using the window as a launchpad. Joseph chases after him, first on foot, then by car, desperately trying to catch the little green baby. Mot flies into a hotel and zooms into a lecture hall where a UFO skeptic dismisses alien stories until the green Martian baby suddenly appears before him. The skeptic’s disbelief turns to shock as he begins to laugh and then cry, overwhelmed by the tiny alien. Joseph arrives just in time to see Mot flying out another window. He attempts to grab the spaceship but fails, falling out of the window himself while calling for the return of Yob, the original baby now on Mars.
Mot, meanwhile, is taken aboard a mother ship, disappearing into the sky. The scene then shifts back to the hospital waiting room, where it is revealed that all of these events were part of a vivid dream Joseph experienced. He wakes up to see the normal human baby being rolled into the nursery, and he relieves himself with a whistle of relief. As a final humorous detail, the camera zooms in on the baby’s wristband, which bears the name "YOB," confirming that the switch was just a fantastical dream.
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