McCoy calls Kirk to sickbay and informs him that the ship's Chief Medical Officer (himself) has contracted an incurable fatal disease called xenopolycythemia and has only one year to live. However, McCoy assures Kirk that he will still be able to do his job until the end.
Suddenly, the Enterprise is attacked by chemically fused missiles and forced to destroy them. The Enterprise then diverts and determines their point of origin to be an asteroid 200 km in diameter. The asteroid is actually a nuclear-powered spaceship on a collision course with planet Daran V (population 3 billion 724 million, 396 days until impact). Spock detects no life forms, so a landing party consisting of Kirk, McCoy, and Spock beams down. While exploring a set of pink cylinders, the landing party is attacked by plaid-clad soldiers. McCoy is knocked out, but proves to be none the worse for wear.
After informing the ``Oracle'' which advises the priestess-leader Natira that they come in peace, the Oracle treats them to an electrical shock to show them what it is like to be enemies. The inhabitants of the world called Yonada do not know that they are on a spaceship, except for one old man who had climbed a mountain when he was young. He gives herbs to the threesome after their electrical experience, but confides to them that things are not as they seem, ``For the world is hollow and I have touched the sky.'' After uttering this, the oracle punishes the old man with death by means of a subcutaneous ``instrument of obedience.''
Spock recognizes Fabrini writing on wall of the oracle chamber and a symbol showing 8 planets, as in the Fabrini system. (The Fabrini appear for the first time in the Star Trek: The Next Generation series.) The sun of this system supernovaed 10,000 years ago, so the inhabitants of Yonada appear to be their descendants. The oracle tells Natira that the people will reach a world of plenty ``soon'' (which turns out to be 390 days after the appropriate course correction). This is foretold in the book containing the knowledge of their creators. Kirk and company must convince the inhabitants to alter the ship's course before it eliminates the 3+ billion inhabitants of Daran V. In the meantime, the priestess Natira governing the asteroid/spaceship, has asked McCoy to become her mate, despite the fact that McCoy has only one year to live. McCoy agrees and permits insertion of an instrument of obedience. However, Spock and Kirk commit sacrilege by breaking into the Oracle chamber. Luckily, Natira allows them to return at McCoy's request.
Bones finds out about the book and tries to tell Kirk (who has returned to the Enterprise) about it by communicator, but is incapacitated by his instrument of obedience. Kirk and Spock beam down, and Spock removes the gizmo. They tell Natira that her planet is actually a spaceship, and she goes to ask the Oracle for the truth. She is punished and nearly killed, at which point McCoy removes her instrument of obedience. When Kirk and Spock try to access the book by pressing the three lower planets on the left side, the Oracle heats the chamber to incandescence. However, Kirk is able to reach the book. He finds instructions on how to access the controls by applying pressure to the center of the Oracle until it slides aside. They disable the Oracle and put Yonada back on course. They also discover databanks of the Fabrini containing a great deal of medical knowledge, including the cure for McCoy's xenopolycythemia. McCoy decides to return to the Enterprise, but hopes to meet Natira again when the ship reaches the new planet.