Does Odysseus attain immortality

Odysseus is offered immortality and eternal youth by the goddess Calypso if he would stay with her, but Odysseus is drawn by his faithfulness to his wife to refuse Calypso.

Odysseus strives to attain immortality, not by living forever, but by gaining a reputation that will have bards telling of his achievements. In this since, Odysseus achieves total immortality. In the since that he lives forever, alas, Odysseus is out of luck. Apparently between the giant cyclops and the feasting he forgot to look for the fountain of youth.