Both stories were hugely important cultural artifacts in antiquity. Every well-educated person in the Greco-Roman world, and most uneducated people, would know these stories well.

Many Greek city-states had multiple yearly festivals where performers recited and acted out the Iliad and the Odyssey. I'm not sure if Roman cities had the same Homeric festival tradition, but there were clearly recitations and plays being put on.

These stories were so well known that orators and philosophers constantly referenced them in relation to their topics with the assumption that everyone would know what they were talking about.

For instance, the Stoic Philosopher Epictetus makes numerous references to the Iliad as he explains ethical matters in Discourses.

He makes a philosophical point, and sort of jokes, that the Spartan King Menelaus should have realized that he'd gained a great boon when the beautiful but faithless Helen ran away:

“If an impression, then, had prompted Menelaus to feel that it was a gain to be deprived of such a wife, what would have come about? Not only the Iliad would have been lost, but the Odyssey too!” (Discourses I.29-13)

Helen's seducer/abductor is also analyzed:

“Did Paris suffer his great disaster when the Greeks arrived and ravaged Troy, and when his brothers perished? Not at all, … his true undoing was when he lost his sense of shame, his loyalty, his respect for the laws of hospitality, his decency.” (Discourses, I.29-22.23)

What about the famous rage of Achilles? Epictetus had thoughts on that too:

“And when did Achilles come to grief? When Patroclus died? Far from it. But rather, when he himself yielded to anger, when he wept over a young girl, when he forgot that he was there, not to acquire mistresses, but to make war. These are the ways in which human beings are brought to grief, this is the siege, this the razing of the citadel, when right judgements are overturned, when they are destroyed.” (Discourses I.29-24-25)

Even if you just wanted to enjoy popular Roman culture, you'd need familiarity with the Iliad and the Odyssey.

Seneca the Younger was perhaps the most popular Roman playwright of his day.

One of his plays was titled "Agamemnon," and tells the story of Agamemnon, the great king of Mycenae, after he returns home from the Trojan War following the events of the Iliad. (Spoiler: his wife is pissed because he sacrificed their daughter to get favorable winds for his fleet, and he's brought a new lady friend back with him, and his wife has been having an affair, so his wife kills him.).

Another one of his plays, Troades, or "The Trojan Women," follows the fate of the women of Troy after the city has been sacked and their men are killed.

Alexander the Great, the famous conqueror, idealized Achilles and slept with a copy of the Iliad under his pillow. When he took his army into Asia Minor to conquer the Persian Empire, he stopped off at the ruins of Troy to look around. Apparently it was a well-known tourist attraction.

Between political speeches, philosophy, books, and plays, you'd be hard-pressed to go a week without hearing a reference to the Iliad or the Odyssey.

There probably is no cultural artifact in the modern western world that possesses anywhere near the widespread familiarity that the Greeks and Romans had with the works of Homer. In the 19th century, we might have said the Bible, but huge swaths of modern westerners don't know that book very well anymore.