The Enterprise is en route to Ariannus to deal with a lethal bacterial invasion when they detect a shuttlecraft missing from Starbase 4. It's traveling erratically and leaking atmosphere, and Spock detects one injured life form aboard. They bring the shuttlecraft aboard and a figure staggers out: solid black on the left side of his body and chalk white on the left.
McCoy tends to the individual and while they can't account for his pigmentation, he is human enough to provide medical assistance. The alien, Lokai, claims he merely borrowed the shuttlecraft and identifies himself from the planet Cheron, in an uncharted section of the planet. Lokai refuses to answer any other questions and Kirk gets word that they've made contact with an approaching ship. However, it is an invisible scout vessel and approaches on a collision course. They try to evade but the ship disappears… and another black-and-white individual materializes on the bridge. However, he's solid black on the right side of his body and chalk right on the left. He identifies himself as Bele of Cheron and reports that his ship disintegrated after a long pursuit: 50,000 years. He lays claim to Lokai, saying he's a political prisoner and Bele is an officer of the Commission of Political Traitors.
Bele asks to be taken to confront Lokai and taunts him, while Lokai refuses to go back and claims to be oppressed. Bele asserts that he and his kind freed them but they rebelled and tried to destroy their oppressors. Lokai demands sanctuary and Bele insists they return the prisoner to Cheron, but Kirk says they have no extradition treatment. Bele insists they take him to Cheron but Kirk refuses until they complete their mission to Ariannus. Bele doesn't believe it's satisfactory but seems to accept Kirk's authority.
Kirk is called to the bridge and informed that they are off-course. They are unable to override and Bele arrives on the bridge to inform them he's taken command of the ship. Lokai storms up to the bridge as well. Both aliens have personal shields that protect them against phasers and physical contact. When Bele refuses to yield control, Kirk begins a countdown to self destruct with Spock and Scotty confirming. Just before the final five seconds, Bele concedes but puts the ship into circles. He asks Kirk to take them both to Cheron but the captain refuses to deal and Bele releases the ship so it can continue on to Ariannus. Kirk gives them unfettered access to the ship but warns them he'll tolerate no interference.
While Lokai talks to the crew and tries to convince them to his cause, Bele has a drink with Kirk and asserts his confidence that Starfleet will find in his favor and turn Lokai over to him for extradition. They get word from Starfleet that Lokai must be brought back to Starbase 4 and then will be released for extradition later. Bele is displeased, believing the crew has been duped and asserting that Lokai's people are inferior because they're black and white on opposite sides.
The Enterprise arrives at Ariannus and completes the decontamination procedure. They leave orbit only to find themselves locked back on course for Cheron. Bele demonstrates that he's burned out the guidance and self-destruct computer circuits. Lokai storms onto the bridge and attacks Bele. Their fields interact and threaten to destroy the bridge. Kirk appeals to both of them and Bele yields control of the ship. They are within sensor range of Cheron and scan the planet, discovering that everyone on the planet is dead and the cities are in ruins. The two of them attack each other and refuse to listen to Kirk's plea that they give up their hate. Lokai storms off the bridge and Bele chases after them. They run through the ship to the transporter room and first Lokai and then Bele beam down to the planet to continue their struggle. The Enterprise has no choice but to leave them to it and departs.