The Federation starship Enterprise chases the stolen space cruiser Aurora, which overloads its engines during the escape attempt. Nevertheless, Aurora's six passengers are safely transported to the Enterprise just before the cruiser's destruction. The group includes:
Tongo Rad, son of the Catullan ambassador.
Irina Galliulin, a friend of Ensign Chekov.
Dr. Sevrin is a renowned researcher in electronics, acoustics, and communications.
Adam, a musician.
Two other women.

When questioned by Captain Kirk, Dr. Sevrin reveals they are seeking the mythical planet "Eden." The group is uncooperative but respects First Officer Spock, who is familiar with their social movement. After a medical examination, Sevrin is found to be an asymptomatic bacterium carrier and quarantined.

After a music concert featuring Spock, the group plans to hijack the Enterprise and take over Auxiliary Control, directing the ship to Eden. Upon arrival, Sevrin incapacitates the Enterprise crew using an ultrasonic frequency.

When the crew awakens, they learn Sevrin and his followers have stolen a shuttlecraft. Kirk disables Sevrin's sonic device, and he, Spock, Dr. McCoy, and Chekov search for the group on the planet. The planet's surface is beautiful but dangerous, with all plant life secreting a potent acid. They find Adam dead from poisonous fruit and the others injured by the acidic plants. Sevrin, in denial, consumes a toxic fruit and dies.

Back on the Enterprise, Irina bids farewell to Chekov. Spock encourages her and her friends not to abandon their quest for Eden, believing they will either discover it or create their utopia.