In Homer's The Odyssey, Book Nine, it is possible to see Odysseus as an unreliable narrator for two reasons. (This is for the sake of argument.)
When Odysseus and his men land on the Island of the Cyclopses, they meet Polyphemus, a Cyclops who cares for sheep. Polyphemus, in a very inhospitable way (something Homer and his peers would have found reprehensible), imprisons his guests, and then starts eating them. It is only after blinding Polyphemus that Odysseus and the remainder of his men are able to escape. (He also tricks the beastly creature into making a fool of himself when he goes to his neighbors for help.) However, looking carefully over the story, the reader may note that Odysseus and his men go into the cave and take from the giant before he attacks them. Perhaps had they been invited, Polyphemus might have reacted differently.
Because of how the Cyclops reacts to Odysseus, the hero becomes extremely angry and wants revenge. We get only Odysseus' side of the story, and a sense that the creature captures and kills members of the crew for no other reason than his naturally violent demeanor.
The part of the story that might make Odysseus seem an especially unreliable narrator is when he becomes full of himself and acts in a disgraceful way (considering he is a hero). Odysseus makes fun of Polyphemus when he attacks Odysseus and his ship—he taunts the giant. This is unbecoming behavior, and it gives the audience a sense that he is emotionally compromised and may not be as objective as one would hope him to be, thereby not delivering a completely reliable account of what transpired on the island.