The USS Enterprise discovers the abandoned SS Botany Bay left floating in the void of space. A team of Captain Kirk, Doctor Leonard McCoy, Chief Engineer Montgomery Scott, and Lieutenant Marla McGivers transports to the merchant ship. The group discovers a cargo of 84 human passengers, 72 of them in a state of stasis after almost two hundred years, while the other 12 passed away during the voyage. McGivers recognizes the group leader, who is returned to the Enterprise for medical examination after reviving.

Kirk has the Botany Bay taken in tow with a tractor beam, and the Enterprise sets a course for Starbase 12. In the ship's sickbay, the group's leader regains consciousness and introduces himself as "Khan." McGivers is astonished to meet Khan, a living reminder of the 20th century, a subject of her study. Spock, the First Officer, discovers their guest is Khan Noonien Singh, a product of a selective breeding program in the 20th century to create perfect humans. Unfortunately, the genetically modified superhumans instead became tyrants and, during the 1990s, dominated more than a third of the planet in the Eugenics Wars. Nevertheless, all the command-crew members admired Khan since he was a fair dictator, with no massacres under his rule and no war until he was attacked, and he had a great vision to bring humanity together. Later on, Kirk asks Khan why he left Earth when his leadership was needed the most; and Khan answers that the superhumans were trying to offer the world "order" but were denied.

After Khan was under guard in his quarters, McGivers was sent to update him on recent developments. Leveraging McGivers' interest in him, Khan informed her of his plan to find a planet that would permit him to lead it and that he needed her assistance to gain control of the Enterprise. Reluctantly, she acceded and beamed him to Botany Bay, where he awakened the rest of his people. They then returned to the Enterprise and assumed command of the vessel, except they could not pilot it. Khan then confined Kirk in a decompression chamber and threatened to slowly asphyxiate all of them unless at least one of Kirk's crew members agreed to help him, assuring that they would be treated well if they decided to join him and argued that bettering humanity was a higher priority than improving machines. However, none of the crew complied, and Khan was left flabbergasted.

After feeling remorseful, McGivers rescues Kirk from the chamber. Kirk and Spock spread an anesthetic gas throughout the ship to hinder Khan and his team. Despite the gas, Khan escapes and goes to Engineering, intending to destroy the Enterprise. Kirk meets him there, and they battle until Kirk knocks Khan out.

Kirk arranges a tribunal to decide between Khan and his people. Unexpectedly, Kirk cancels all charges, deciding that sending them to a penal colony would be a poor use of their potential. Instead, he proposes to Khan that he either live in the present day as an ordinary person or inhabit and manage Ceti Alpha V, a deserted, flourishing planet that he compares to the Botany Bay Colony on Earth. Khan consents to Kirk's suggestion, quoting Milton's Paradise Lost. Instead of a court-martial for McGivers, Kirk enables her to accompany Khan. Spock remarks that seeing what Khan will make of Ceti Alpha V in a century would be fascinating.