Reading the Odyssey reveals a world far more existentially brutal than that of the modern West. Its attitudes towards women are particularly unsettling.
There is a double standard at play in the text, by which Odysseus is allowed to be a philanderer, while Penelope must remain faithful to her husband. However, this is actually just the tip of a much deeper iceberg. Consider, for example, in book nine where Odysseus discusses the sack of Ismarus. In the Fagles translation, Odysseus states:
There I sacked the city, killed the men, but as for the wives and plunder, that rich haul we dragged away from the place—we shared it round so no one, not on my account, would go deprived of his fair share of spoils.
Note how the women are ultimately reduced to the status of objects and possession, removed of all humanity or dignity. Note also the lack of shame or guilt on Odysseus's part. Quite on the contrary, Odysseus is boasting about this act of intense victimization and violation. With this in mind, it might also be useful to remember the Iliad, which begins with the taking of women as sexual slaves.
In addition, you can also point towards the events of book 22, when Odysseus, after killing the suitors, proceeds to kill the disloyal female servants (some translators translate the word as "slaves"), referring to them (in the Fagles translation) as "you sluts—the suitors' whores." There is no discussion or consideration taken into the disparity of power between the servant women and the suitors, and the women are given no chance to defend themselves. Thus, the poem's treatment of women goes beyond the problem of marital infidelity, to reveal world brutally hostile towards them.
In addition, I think this point towards an even more existential problem, by which the world itself was far more casually brutal than the modern world today. Note Penelope's relative helplessness against the suitors, who seek to pressure and coerce her into choosing a new husband after Odysseus has disappeared. Ultimately, this comes down to a matter of force: with her husband gone, Penelope is vulnerable. With both her and her household weaker than they are, the suitors are thus free to impose upon her and her son. Note too that Odysseus's great triumph represents a reversal on this same theme: Odysseus is stronger than the suitors, and thus slaughters the lot of them in reprisal for their insult. Taken as a whole, however, there does not seem to be a modern sense of due process or rule of law. Certain cultural expectations hold true and remain in force (hospitality, for example, emerges as one of the core cultural values that binds the poem together), but ultimately, the world of the Odyssey seems to be largely dictated by power and use of force.