When Kirk, Spock, and Bones beam down to a small planet in the Omega system to obtain ryetalyn to counter the epidemic of Rigellian fever sweeping the Enterprise, they detect a life form (despite the fact that ship's sensors had indicated it to be uninhabited). They are then attacked by a droid called M4, which is subsequently called off by its owner, Mr. Flint. Mr. Flint demands that the landing party leave the planet immediately, and threatens to kill them if they do not. Kirk then calls his bluff by ordering Scotty to lock phasers on their position so that an attack on the landing party will result in the death of all of them. When McCoy tells Flint that the crew of the Enterprise is suffering from a disease which has effects similar to bubonic plague, Flint recalls the progress of that disease in Constantinople in the summer of 1334. He then permits the crew two hours on his planet, and offers M4's services to collect the ryetalyn. He then invites them to his home.
In Flint's home, McCoy is impressed to find a Shakespeare first edition, Gutenberg Bible, and creation lithographs by Terra Nullus of Centaurus 7. Spock then examines the extensive collection of unknown Da Vinci paintings, but his tricorder reveals the materials used to be of contemporary origin.
Meanwhile, Flint has consulted with his protege Rayna Kontec, who begs him him to allow the landing party to stay a bit longer. Flint then promises to have M4 process the ryetalyn, and introduces Rayna to the landing party, who are smitten with her. She, however, seems most interested in discussing field density and its relationship to gravity phenomena with Spock. Flint claims Rayna's parents were killed in an accident while in his employ and that he has raised her. He also says that Rayna possesses the equivalent of 17 university degrees in the arts and sciences.
While Kirk plays billiards with Rayna, Spock discovers a waltz on the piano and plays it while Kirk and Rayna dance. Spock is greatly surprised (for a Vulcan, at least), when he recognizes the waltz as an unknown work of Johannes Brahms written in his own hand.
In Flint's lab, McCoy discovers that the ryetalyn is contaminated with iridium at nearly 1 part per thousand, rendering it inert and useless. While Flint goes to collect uncontaminated ryetalyn, Kirk takes the opportunity to romance Rayna, with whom he has fallen in love. While Kirk is thus engaged, M4 returns and prepares to attack Kirk, who is saved when Spock destroys M4 with a phaser. Flint explains that M4 misunderstood Kirk's actions as an attack and was attempting to defend Rayna. Just as Kirk is beginning to feel comfortable after M4's demise, a replacement M4 created by Flint makes its appearance.
Kirk contacts the Enterprise to try to find information on Flint and Rayna. However, the only information Uhura can provide is that the planet was purchased 30 years ago by a wealthy financier and recluse named Mr. Brack. Spock finds another piece of the puzzle when he surreptitiously performs a tricorder scan and discovers that Flint is 6000 years old.
When Rayna goes to say goodbye to Kirk, Kirk urges her to come away with him. Meanwhile, the ryetalyn has also disappeared. When Kirk follows tricorder readings past a sliding panel, they find a chamber containing a series of android Raynas created by Flint to act as companions. Flint then admits to having been born in Mesopotamia as the soldier Acarin in the year 3834 BC. After falling in battle, he learned that he was immortal. In his lifetime, he was also Abramson, Alexander, Brahms, Da Vinci, Lazarus, Merlin, Methuselah, Reginald Pollock (from the 20th century), Shakespeare, Solomon, the painter Sten (from Marcus II), and many more people. Because of his immortality, he created a mate who would not age and die. Having learned Flint's secret, Flint refuses to let the landing party go. When Kirk tries to warn the Enterprise to leave the area, Flint suspends the Enterprise as a small tabletop model.
Flint is happy that Kirk has stirred emotions in Rayna and wishes to turn them towards himself. When Kirk refuses to back down, he and Flint fight one another. As a result of the conflict of Kirk wanting her to come away and Flint wanting her to stay, Rayna is caught in a logic loop because she is unable to hurt Kirk or Flint. This causes her to die. Kirk is devastated, but obtains uncontaminated ryetalyn from Flint and returns the the Enterprise.
On the Enterprise, Bones discovers that Flint is dying, having been removed from the Earth whose complex ``fields'' were the source of his immortality. Kirk's emotional trauma at the loss of Rayna is healed only when Spock uses a Vulcan mind meld and makes him ``forget.''