Stardate 3468.1: Lt. Carolyn Palamas submits her report on Pollux 5 to Captain Kirk, and Scotty is clearly interested in her. McCoy notes she'll probably find the right man some day and leave the Service. The ship enters orbit around Pollux 4 but a giant glowing energy-hand moves toward them from the planet's surface. Before they can pull away, the hand grabs them.
The crew tries to break free without success and a humanoid male face appears in space before them. The man speaks to them, greeting them as children who have come a long way. When Kirk demands they be released, the figure applies pressure until they are forced to surrender. He then invites Kirk and his officers down but tells them to leave Spock behind. Kirk takes McCoy, Scotty, Chekov, and Palamas down to the surface where they find an ancient Greek temple. The man, wearing a toga, greets them and welcomes them as his children. He talks of Earth then identifies himself as the Greek God Apollo. He shuts down their communicators and demands their worship, and when they refuse he grows to giant size, towering over them.
While Spock attempts to locate them on the planet, Apollo disappears and Kirk has Palamas tell them of the legends of Apollo. They look for the source of Apollo's power while Kirk conjectures that their host actually is Apollo. Apollo reappears and again demands their worship. He offers Palamas the role of a goddess and when Scotty tries to attack him, destroys the phaser with a lightning bolt and fuses the other ones into uselessness. He transforms Palamas' uniform into a toga and sends Scotty flying when he tries to interfere, then teleports away. They wake up Scotty and Kirk tells the engineer to find the power source and keep his temper in check. Kirk then speculates that Apollo and his fellows might have been aliens who landed in Greek 5,000 years ago and frightened the locals, taking the role of gods.
While Spock and Uhura work to restore communications, Apollo explains to Carolyn that on Greek they eventually lost the worship of the Greeks and managed to come back to their homeworld of Pollux 4. The other "gods" dissipated themselves upon the wind but Apollo waited, confident they would travel to the stars.
Chekov and Scotty can't pinpoint the energy but Chekov suggests Apollo is tapping the power directly through his power. McCoy has identified an extra organ in Apollo's chest, but before they can discuss it further, Apollo returns alone. Scotty attacks him and is blasted back again in deep shock. When Kirk defies Apollo, the "god" starts to telekinetically chokes him. However, Apollo seems weakened and then teleports away. Chekov notices and reports his findings as Scotty and Kirk revive. They figure that between maintaining the force field on the Enterprise and firing the occasional energy blast on the planet, Apollo has gone to recharge his energy cells. They figure if they all jump him, he'll drain his energy on one of them and the other three can defeat him.
Spock is preparing to use m-rays to fire through the force field/"hand." Apollo returns with Carolyn and demands their worship, and they respond by mocking him. However, Carolyn intervenes and persuades him to give them one more chance, and he reconsiders, telling them to dismantle the ship for the supplies they need and prepare to worship him. He disappears with Carolyn again and he acknowledges her wisdom and they kiss.
Carolyn returns to the landing party and tries to convince them to surrender, but Kirk tries to turn her against Apollo to deprive him of the worship he needs She is summoned back and acknowledges her duty before teleporting away. Uhura manages to establish contact with the landing party and Spock informs them he has pinpointed the power source: Apollo's temple. The Enterprise prepares to open fire but Kirk holds off until they know where Carolyn is.
In the woods, Carolyn starts treating Apollo as a specimen, dismissing his romantic interest. She walks off as he flies into a fury and summons a thunderstorm and grows to giant size to go after her. Kirk orders Spock to fire the phasers as they run for cover. Apollo fires lightning bolts at the Enterprise. The phasers reduce the temple to slag and Apollo is left all but powerless. Kirk regretfully informs Apollo that they've outgrown him, and Carolyn can only weep as Apollo grows to giant size one more time, then throws himself upon the wind, vanishing. Kirk is left to wonder if perhaps they might have gathered a few laurel leaves.