What happens to Greek society when Odysseus leaves his home in Ithaca to go on his epic journey?
The most noticeable change to Greek society in terms of Homer's story The Odyssey is what occurs in Ithaca, specifically with regard to Odysseus' home.
When Odysseus travels to Troy to fight in the Trojan War, he is gone ten years. As he starts his voyage home, he visits extraordinary places, is held captive for several years, and angers a god or two. His journey home stretches out for ten more years.
Meanwhile, things are happening at his home—stemming from the ignoble behavior of noblemen of Ithaca.
In Book Two, in front of an Assembly of the people of Ithaca, Telemachus, son of Odysseus, describes two disasters that have befallen his home. First, Odysseus has not returned from the war. Second, a group of suitors—sons of the most prominent of Ithaca's citizens—have taken over his home.
Like vultures sensing something dead, suitors for Penelope's hand have descended upon her in the absence of her husband, and taken up residence in Odysseus and Penelope's home. The men believe that Odysseus is dead.
The suitors break the laws of hospitality not only by coming uninvited, but also by staying—and refusing to leave. Telemachus is really too young and too inexperienced to throw these unwanted guests out. He was born just after Odysseus left, so he is not yet twenty. There are those in the household who are loyal to Penelope, but others have welcomed these enemies of Odysseus who want nothing other than to take possession not only of his home and his belongings, but to take Penelope to wife as well.
So Telemachus goes to the Assembly to ask their help. None of the men of the council come to his aid, but several of the suitors verbally attack Telemachus. Antinous accuses Penelope of leading all of them on with no intent of marrying any of them.
Telemachus notes that to marry Penelope to another man, he would have to send her home to her father, but he believes that this would be unjust, and that the gods would punish him. He tells Antinous that if he is unhappy at Odysseus' home, he should leave. If he does not, he will have to reckon with the gods:
...If you choose to take offense at this, leave the house and feast elsewhere at one another's houses at your own cost turn and turn about. If, on the other hand, you elect to persist in spunging upon one man, heaven help me, but Zeus shall reckon with you in full, and when you fall in my father's house there shall be no man to avenge you.
Two eagles from the heavens fly into the room, fight and then depart. Halitherses, the prophet, sees the eagles as an omen and delivers a prophecy. He warns the Assembly that Odysseus will return; the suitors must leave Odysseus' home willingly; and, that the suitors are not the only ones who need to listen to him.
...indeed [Odysseus] is close at hand to deal out death and destruction, not on [the suitors] alone, but on many another of us who live in Ithaca. Let us then be wise in time, and put a stop to this wickedness before he comes.
Another of the suitors, Eurymachus, threatens the prophet (that he should go home or his children will "miss" him:
Go home, old man, and prophesy to your own children, or it may be worse for them.
Eurymachus tells him that the eagles mean nothing—Odysseus is dead. In all, no one is willing to help Telemachus. The noble society and its laws have declined in Odysseus' absence, evident not only in how the suitors act, but in the way the men of the council fail to stop the wicked behavior of the suitors.