Odysseus refuses immortality because he wants to return home. Calypso, the nymph who keeps him on her island, offers him a life without old age or death beside her, and he prefers to return to Ithaca, to Penelope, to his son and his father, even knowing that he will remain mortal and may still suffer greatly on the way.
The scene is in Book V. After seven years on the island, Odysseus spends his days weeping on the beach, with his eyes on the sea, and his nights with the nymph, "out of constraint." When Hermes brings Zeus's order to release him, Calypso complains that the gods are "more envious than all" and cannot bear that a goddess sleep openly with a mortal. She yields, but still tries to persuade him: if he knew the sufferings that fate prepares for him until he reaches his homeland, he would stay with her and be immortal.
Odysseus's answer is courteous and firm. He admits that the prudent Penelope "is inferior to you in appearance and stature," because she is mortal and Calypso is immortal and ageless, "but still, I want and desire every day to return home and see the day of homecoming." And he accepts the price: if a god wrecks him at sea, he will bear it, for he has already suffered much on the waves and in war. Calypso, in turn, gives him an axe, tools, and provisions, and Odysseus builds the raft.
The poem leaves the choice without comment, but reinforces it by contrast. In Book XI, the soul of Achilles says he would rather be a farm laborer serving a poor man than rule over all the dead; to live, even poorly, is worth more than glory without life. Odysseus's refusal shows the same value by another route: what gives meaning to life, in the Odyssey, is not duration nor splendor, but the house, the family, and the land where one is known by name.