Ship's surgeon Doctor Leonard McCoy has discovered that he has contracted a rare disease called xenopolycythemia and only has about a year to live. The Enterprise comes under attack from some old-fashioned missiles that have been launched towards it from a large asteroid. The asteroid has been under observation as it is on a collision course with the inhabited planet Daran V. The starship easily deals with the threat and, as the asteroid appears to have a breathable atmosphere, Kirk, Spock and McCoy beam down to investigate further.

The Starfleet officers are almost immediately captured by some armed men and a beautiful woman, who appear from out of strange metal cylinders protruding from the surface of the asteroid. The woman announces herself as Natira, the High Priestess of Yonada, and orders the three prisoners to be taken inside the asteroid. They are led into a room decorated with strange symbols where Natira speaks with her 'god', the Oracle who stuns the Enterprise men with a power beam. Kirk and Spock awake in a luxurious room but McCoy, weakened by his disease, takes longer to recover which is of concern to Natira who wants him for her 'mate'.

She tends to McCoy while Spock and Kirk are allowed to wander around, discovering that the asteroid is hollow and supports a self-contained world and its population. They get into a conversation with an old man who realises that they are different. He tells them that the people do not know they are living in a hollow sphere and how, when he was young, he climbed a high 'mountain' and discovered the fact for himself 'for the world is hollow and I have touched the sky'. Suddenly, something under the skin of his temple starts glowing and he falls dead at their feet.

Kirk and Spock make their way back to the oracle room where Spock learns that the inscriptions in the stone walls were made by a race called the Fabrini who created the asteroid 'spaceship' in order to save their race when their sun went nova ten thousand years ago. They are discovered and removed before he can examine them further. McCoy has grown fond of Natira and, as he has only a year to live, decides to stay with her on Yonada, persuading her to allow Kirk and Spock to return to the Enterprise. In order to be allowed to stay on Yonada he agrees to allow the Oracle to insert a subcutaneous 'instrument of obedience' in his temple.

After the ceremony McCoy is allowed access to the 'Book of the People' which, he is amazed to find, contains illustrations of star systems. He attempts to contact Kirk aboard the Enterprise to let him know of his discovery but he is prevented from completing the call by his 'instrument of obedience' which is activated by the Oracle. Kirk and Spock return to Yonada, removing the alien device from his skin and explaining to Natira that her 'world' is actually a giant spaceship. She queries the Oracle about this but is punished for her blasphemy. The obedience device is removed from her skin and the walls of the oracle chamber start to heat to a red glow.

Kirk examines the 'Book of the People' and discovers how to access the control room where he and Spock manage to disengage the Oracle's defence systems. A fault in the asteroid's systems is located and repaired and Yonada is put back onto its intended course. The knowledge contained in the Fabrini book also furnishes McCoy with a cure for his disease. Sorrowfully, McCoy and Natira agree to part, he to return to his rightful place aboard the Enterprise and she to lead her people to their new home planet which Yonada will reach within a few months.