The crew of the U.S.S. Enterprise has become infected with Rigelian fever and the outbreak is reaching epidemic proportions. Three crewmen have already died and twenty-three more are in a serious condition. The only known antidote for the illness is a substance called ryetalyn, deposits of which the Enterprise has managed to locate on an uncharted planet. Kirk, Spock and McCoy beam down to collect the quantities of the substance that the doctor will need to cure the crew.

The landing party comes under attack by a hovering robot unit. When they try to defend themselves they find that their phasers have been disabled and are only saved by the appearance of a humanoid male who orders the machine to halt its attack. The man identifies himself as Flint, the owner of the planet, and requests them to leave immediately. However, after hearing of their problem he invites the three of them to be his guests while he arranges for his
M-4 robot unit to collect the ryetalyn for them.

His home is a virtual palace filled with original art and antiques with which Spock becomes fascinated as many of them are unknown works by famous artists. Kirk is equally fascinated, but the subject of his attentions is a beautiful young woman called Rayna Kapec who, he is told, has been Flint's ward ever since her parents died in an accident. The first consignment of ryetalyn collected by M-4 is examined by McCoy and proves to be contaminated and useless, delaying their return to the Enterprise. Kirk spends much of the time with Rayna who he is falling in love with.

He receives a call from McCoy, who is in Flint's extensive laboratory, telling him that the second consignment of ryetalyn has disappeared. Kirk makes his way to the lab and, while searching for the ryetalyn, enters a room where he discovers earlier models of Rayna who is, in fact, an android. Flint appears in the lab and explains his history to them. He was born on Earth in the region known as Mesopotamia in 3834 B.C.E. As a soldier called Akharin he received what should have been a fatal wound in a battle but did not die and discovered that he was immortal. He has lived throughout the centuries under such names as Leonardo DaVinci, Brahms and many others.

The art and original manuscripts that have fascinated Spock had all been created by himself at various times. Flint tells them that he created Rayna to be an immortal companion for himself and had taken the opportunity of using Kirk to rouse the latent emotions in the android. He also tells them that, in order to keep his existence a secret from the rest of the galaxy, he has to prevent the Enterprise from leaving and has invented a device that will shrink the starship and keep the crew in a state of suspended animation. Kirk attacks Flint but, as they fight, Rayna enters the room and sees them, begging them to stop. Her newly found emotions cannot handle the contradictory feelings that she has for both men, causing her to overload and 'die', leaving both Kirk and Flint stricken with grief.

The landing party is allowed to return to the Enterprise with the ryetalyn after vowing not to reveal Flint's existence and the crew is cured. On examining medical scans of Flint McCoy reveals to Kirk that the man is dying as whatever caused his immortality was dependent on him remaining on Earth. Kirk is still deeply troubled by the 'death' of Rayna as he had fallen in love with her. Spock makes the decision to help his friend and, unknown to Kirk, eases his pain while he sleeps by carrying out a mind-meld to help him to forget Rayna.