The USS Enterprise, a ship from the Federation, answers an emergency signal from an uncharted world. An away team is beamed down to the planet to find the origin and immediately encounters two human-like figures, Rojan and Kelinda, who are from the Kelvan Empire; they incapacitate Captain Kirk and his team and demand him to yield the Enterprise. Rojan explains that the Kelvans are from the Andromeda Galaxy and have come to the Milky Way looking for planets to take over.

Their vessel was destroyed by a negative energy barrier on the galaxy's edge, leaving them in need of a ship. They planned to take the Enterprise for the 300-year journey home. However, three Kelvans then beamed aboard the Enterprise and took control of the ship.

Rojan commanded the landing party to go to a cave that Kelinda was monitoring. Mr. Spock used his Vulcan mental power to draw Kelinda into the cave, at which point Kirk rendered her unconscious and seized her control belt. Nevertheless, their freedom was short-lived. As a punishment, Rojan commanded Kelvan Hanar to activate his belt, resulting in two landing team members being transformed into small cubes of a chalk-like material. Rojan then picked up the cubes and crushed one, killing Yeoman Thompson. He then changed the other cube (Lt. Shea) back into human form.

Spock told Kirk about his encounter with Kelinda, explaining that the Kelvans are not human-like but have taken on human form for practicality. They are enormous and have hundreds of tentacles. Spock had been forced out of the mind meld because the Kelvans were so foreign and had such influential minds.

Kirk and Spock attempted to modify McCoy's neural scanning equipment to make a defense against Kelvan's paralysis field. Spock faked being ill by going into a trance-like state to get beamed up to the ship. Afterward, the Kelvans took the rest of the landing team and themselves to the ship.

Spock concludes that breaking into the paralyzing field projector's outside layer is impossible. As a substitute, he has Scott set up the matter/antimatter system to explode when the barrier is touched if Kirk gives the command. However, Kirk chooses not to do this. After the ship passes through the galactic boundary, the Kelvans convert the crew members apart from Kirk, Scott, Spock, and McCoy into powdery blocks. Spock recalls more details from his mental connection with Kelinda and reveals that the Kelvans, in their original form, have feeble senses and emotions. Still, in their current state, they show normal human reactions. Kirk decides to take advantage of their lack of experience with physical and emotional impacts.

Scott introduces Tomar, an engineer from Kelva, to alcoholic drinks, McCoy injects Hanar with a "supplement" that has a stimulating effect, and Kirk starts flirting with Kelinda, which produces a feeling of jealousy in Rojan. In the end, Kirk can get Rojan to strike him. He points out that Rojan is responding humanly and that his descendants who will reach the Andromeda Galaxy will be considered alien inferiors by the Kelvans. By the time Kirk finishes speaking, Rojan realizes he is right and passes control of the ship to Kirk, who then orders them to return to their galaxy. The planet where Rojan and his people were stranded is suggested as their new residence.