Take a story's temperature by studying its tone. Is it hopeful? Cynical? Snarky? Playful?
Homer writes with all the gravity that you'd pretty much expect when reading about epic heroes and their long dangerous journeys. Odysseus's suffering is endless; Telemachos risks his life to find out news of his father; the god's ponderously debate about what to do to the mortals next, and Penelope wishes she were dead, and not in a melodramatic, I-hate-my-parents way:
So I wish that they who have their homes on Olympos would make me vanish, or sweet-haired Artemis strike me, so that I could meet the Odysseus I long for, even under the hateful earth, and not have to please the mind of an inferior husband. Yet the evil is endurable, when one cries through the days, with heart constantly troubled, yet still is taken by sleep in the nights; for sleep is oblivion of all things, both good and evil, when it has shrouded the eyelids. But now the god has sent the evil dreams thronging upon me. For on this very night there was one who lay by me, like him as he was when he went with the army, so that my own heart was happy. I thought it was no dream, but a waking vision. (20.79-90)
This is serious stuff. But that doesn't mean the epic is all doom and gloom. We get some comic relief along the way. Take this passage, when Circe turns Odysseus's men into pigs:
When she had given them this and they had drunk it down, next thing she struck them with her wand and drove them into her pig pens, and they took on the look of pigs, with the heads and voices and bristles of pigs, but the minds within them stayed as they had been before. So crying they went in, and before them Circe threw down acorns for them to eat, and ilex and cornel buds, such food as pigs who sleep on the ground always feed on. (12.237-43).
That's comedy gold, right there.
Homer even gives us a metaphor for the poem's tone, when he compares Odysseus stringing his bow to a singer tuning his lyre (Book XXI, lines 404-411). The Odyssey is just like that: delicately—perhaps dangerously—poised between harsh and warlike (the bow), and beautiful and enchanting (the lyre).