In response to a distress call, a landing party from the Enterprise has beamed down to a planet's surface but cannot detect any signs of life. Eventually, two readings are picked up by Yeoman Thompson which prove to be a humanoid male and female who approach the party. The man orders Kirk to surrender the Enterprise to him. On Kirk's refusal the alien couple operate belt-mounted controls which leave the landing party paralysed. The aliens, called Rojan and Kalinda, identify themselves as Kelvans from the neighbouring Andromeda galaxy. Rojan tells Kirk that he is now in control and any attempts at resistance will be severely dealt with. He also warns them that they will soon be leaving the galaxy and that they must face the end of their existence, as they know it.

The Enterprise team is released after Kalinda removes their weapons. Rojan continues to explain that they have been sent as a scout party to take over this galaxy as their own is becoming uninhabitable. They need the Enterprise as their own ship was destroyed coming through the barrier surrounding the galaxy and plan to modify the ship's engines to enable them to make the return trip, which will take about 300 years. While he is talking, three other Kelvans materialise aboard the Enterprise and quickly gain control of the ship. One of them, Hanar, beams back down to report to Rojan.

The landing party is incarcerated in a nearby cave, modified to be a prison cell. Spock uses his Vulcan telepathic abilities to lure Kalinda to them where they overpower her and remove her control belt. They manage to get outside but are stopped by Rojan who punishes them by reducing two of their members into small, multifaceted chalk-like blocks. He tells them that the blocks are what remains of the humans after they have been reduced to their basic elements, picks one up and crushes it to pieces in his hand. Hanar restores the undamaged block back into human form but the crushed block, which was Yeoman Thompson, cannot be revitalised.

They are returned to thecell cave where Spock analyses the images he received while in mental contact with Kalinda's mind , telling Kirk that he received the impression of massive creatures with hundreds of limbs and believes that this is the Kelvans' true appearance. He surmises that they have adopted humanoid form in order to be able to use the Enterprise. Spock puts himself into a self-induced trance and McCoy persuades the Kelvans to allow him to take Spock to sickbay on board the Enterprise. The Kelvans eventually transport the rest of the party back aboard and Kirk makes his way to sickbay where he learns that Scotty and Spock have found a means of 'jamming' the power source on the Kelvans' belts.

On making their way to engineering, Spock and Scotty find that they cannot get at the Kelvan machinery that is modifying the ship's engines. Instead, they open the control valves in the matter / antimatter chamber, reporting to Kirk that the ship can be destroyed on his order when it reaches the edge of the galaxy, but Kirk decides against this option. After passing through the galactic barrier, the Kelvans reduce all non-essential personnel to the chalk-like block condition, leaving only Kirk, Scotty, Spock and McCoy to help run essential functions. From a Kelvan called Tomar they discover that the aliens are unused to the human sensations that they have been forced to adopt along with the bodies and Kirk sees an opportunity, ordering the others to find as many ways of 'stimulating' the Kelvan senses as possible.

Scotty gets friendly with Tomar and takes him to his quarters where he plies him with powerful Saurian brandy. McCoy prescribes a course of injections for Hanar after telling him that his human body looks unwell, and Kirk turns his attentions towards Kalinda. He apologises to her for his previous behaviour and demonstrates the 'human' way of saying sorry by kissing her. Kirk quickly leaves when Rojan walk in on the pair of them. Later, while playing chess with Rojan, Spock suggests that his lack of concentration on the game is due to jealousy. Rojan forcefully disagrees but goes to Kalinda and tells her to stop fraternising with Kirk and the other humans, grabbing her violently when she refuses to obey.

Kalinda gets Kirk alone and asks him to kiss her again. Meanwhile, McCoy's course of injections has made Hanar very argumentative and irritable and the Kelvan is confined to quarters by Rojan when his orders are questioned. Scotty enjoys a different sort of success, reducing Tomar to drunken unconsciousness, but he is too drunk himself to leave his quarters. Rojan finds Kirk and Kalinda together in the rec room where Kirk goads him into starting a fight. He fairly easily manages to pin Rojan down, telling him that they are already becoming human and that their descendants will be completely alien to their own kind when they get back to Kelva in 300 years time. Rojan thinks about this and agrees, returning control of the Enterprise and accepting Kirk's offer of Federation help if his people come in peace.