The play begins shortly after Black has saved White, a professor, from committing suicide: He attempted to jump in front of a commuter train, the Sunset Limited of the title. To be clear, the two characters are identified by their respective races rather than their names. As they sit in Black's apartment, Black gradually tries to convince White that life is worth living.
They quickly get into religious territory after White asks Black if he thinks Jesus is present in the room (White's an atheist and finds religion to be ridiculous; Black's a Christian). This expands into a discussion about whether the Bible is the greatest book of all time or not, and proceeds from there.
They talk more about White's suicide attempt and about why White finds life meaningless. White's a kind of "culture junkie," as Black puts it, but he thinks books, art, music and all the things he used to believe in have lost their value. As he now sees it, the world is a place of suffering, and humans only delude themselves into sticking around.
On the flip side, Black's an optimist. Although he's had a rough life—both of his sons are dead, he was in jail for murder, and he's done a lot of bad things—he's much more positive than White. He tries to convince White that the key to happiness is loving your fellow humans and surrendering to a higher power. In his view, "life everlastin" is a condition you can experience in this life by getting in touch with the underlying soul shared by everyone.
A for effort, but White doesn't believe in the central importance of other people—he believes reason and the intellect are the most important things. They're what you need to follow, he thinks.
At one point, they stop to eat dinner, with Black heating up leftovers—some authentic New York fusion food he made himself. This provides a short lull in the conversation before they get back to a more intense argument. After eating, it seems like White's regained some energy—he starts defending his nihilistic outlook with more zeal.
White eventually explains that he sees the world as a "forced labor camp" where the workers—meaning everybody—are gradually selected for execution, one by one. Eventually, Black can't think of any more arguments. White just won't give up on giving up, by his own admission; he thinks life is a howling void, and he just wants death, just wants to stop existing in order to minimize his pain. Finally, Black lets White out the apartment.
As White walks away—apparently to kill himself—Black says he'll return to the station and save him again. Black breaks down, crying, and asks God why he didn't give him the words needed to save White. But he reconfirms his faith and says he'll be true to God's word even if God never speaks to him again. The play ends with Black asking God, "Is that okay?"