- Russia sends a dog called Whyka into space. With the help of a monkey called Buster, Dot rescues the stranded dog only to end up on a planet ruled by fascist Papa Drop who wages war on those who aren't perfectly shaped.
- Dot is watching a portable TV set in the bush with her kangaroo friends Dozey-Face and Grumble-Bones. Through this, they witness a Russian dog called Whyka being launched into space, the mission goes well until Whyka's rocket becomes trapped in orbit. Dot, concerned for Whyka, insists on attempting to rescue the dog despite the other bush animals having their doubts. Dot and Dozey-Face travel to America where Dot infiltrates the Space Lab and finds Buster the monkey who was also due to be launched into space. Noticing Buster's passion for music, Dot trades him her musical watch after demonstrating it in exchange for his spacesuit, she briefly has to imitate him to avoid detection.
The next day, Dot's rocket is launched, she soon finds Whyka's rocket. Dot narrowly manages to rescue Whyka before the dog's rocket explodes, the resulting shock-wave sends Dot's rocket off-course and she crash-lands on the nearest planet. Dot and Whyka find the planet to appear deserted and devastated. Quickly deducing they're not on Earth, an increasingly-afraid Dot and Whyka leave their rocket, intending to explore. The planet's local soldiers, called the Roundies, are alerted to their presence however and quickly capture them. Dot is tied up and gagged and taken to the Roundies' tyrannical leader Papa Drop. The abusive dictator has his adviser Professor Globus examine Dot and gives Whyka to his son Roley. Because Dot obviously is not round, she is taken away for interrogation and finally thrown into the Prison for Squares.
While Dot is a prisoner of the Roundies, she befriends another prisoner called Poley who explains that the Roundies and Squaries once co-existed peacefully and Papa Drop has destroyed all this through his own prejudice and propaganda. Dot is eventually reunited with Whyka whom Roley uses to send messages to Poley. Dot endures several days of torture and slave labor before employing several tactics inspired by Poley's story to trick the Roundies, such as smashing a mirror to distort their reflection and painting separate paths for the prisoners, enabling her to escape.
Dot and Whyka are unable to use their rocket to escape as it is still stuck in the ground where they crash-landed, Dot having been among the prisoners forced to dig it out so Papa Drop can use it to conquer the Universe. They eventually head off in search of The Party, a mysterious entity whom Papa Drop believes to be plotting to overthrow him, the journey sees them encounter many dangers such as quicksand, attacking trees and an active volcano. The Roundies pursue Dot and Whyka and are able to pick up their trail due to Dot losing fragments of her dress.
Dot and Whyka help themselves to a local vegetation called pilly-fruit but their campfire draws the Roundies to them and Dot is bound and gagged once again. The Roundies are suddenly forced to flee when a large red monster called Gorgo appears, leaving Dot and Whyka to seemingly get eaten but Gorgo simply swallows Dot's gag, drawn to the scent of the pilly-fruit. This draws the attention of The Party who rescues Dot and the two of them are able to get the Roundies on their side as they head back to challenge Papa Drop. They proceed to point out what a fraud Papa Drop is and free all the prisoners. The Party then reveals himself to be Roley in disguise and Papa Drop's assistant Inflato pumps the dethroned dictator enough to send him floating away. Roley has Gorgo rescue Dot's rocket, allowing Dot and Whyka to climb aboard. Dot and Whyka proceed to blast off and head back to Earth.
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