Ithaca

A little background and a little logical deduction, using the clues in The Odyssey itself, will help us understand the real issues facing these characters: Penelope, Telemachus and Odysseus.

Ithaca: The War Wife and Child
This is a story replicated millions and millions of time, in all times, including today: this is a story of a family coping with war: war does not only affect soldiers and the cities it ravages but also those left at home.  Consider as well that the Iliad ends with a Priam and Achilles connecting via their common relationship: that of a father and as a son, or what it's like for a father to lose a son, and the Odyssey continues this theme by opening with the inverse: with a son who is missing his father.

Penelope and Greek Women
As the story suggests, Greek women could not own property and they were, for all practical purposes, themselves the "property" of the men in their lives.  As the council tells Telemachus, if Odysseus the husband does not return, Penelope's property should be returned to her father, Iacrius, and then a new husband must be chosen. She does seem to have some say in who she chooses, however. 

Notice that Penelope is never allowed to be alone, and when she meets the suitors she is always veiled.  Nausicaa is treated similarly.  In reality, Greek women in both Homer's time and much later were treated similarly to women in Saudi Arabia or Kuwait (literally thousands of years before the rise of Islam), and these women hardly ever left their family's home and were largely kept as slaves, cut off from the world outside the family compound.

Even in so called "Enlightened" classical Athens (c. 400 BC), women could not hold political power or, likely, property.

The Wife: Beginning with the first pages and continuing throughout, the Odyssey obsesses over Clytemnestra (note her name is the origin of a similar sounding portion of the female sexual anatomy) and Aegisthus's treachery against Agamemnon; we are reminded over and over again what happens when a man leaves his woman alone for too long, at least from the ancient Greek perspective.

Telemachus The Prince In Waiting
Telemachus is a boy-child raised in a home without a father.  This seems to have left him unable to become "a man". Like Hamlet, with his father missing Telemachus fears he will not gain the throne. If Penelope chooses a new husband, and if Telemachus cannot control the Ithacan throne through brute force, he will not become king. 

Cleverly, this story essentially begins where the Iliad left off: with a meditation on the vital relationship between fathers and sons; Priam, whose home is directly ravaged by the war, reminds of this when he pleads Achilles to let go of Hector's body: "Remember your father, Achilles," his plea begins, and so we find that is exactly what Telemachus is doing at the opening of the Odyssey:  remembering his father and the devastation his failure to return will wreak upon his entire city.

Anarchy
So, really, the entire issue of the suitors simply symbolizes the chaos that ensues when a king and all his men go missing for roughly twenty years. Although we see the Greeks had a degree of democracy -- the council attended by Telemachus, and by the Greeks in the Iliad, as well, but when Telemachus convenes the assembly, we learn this is the first time it has gathered since Odysseus left twenty year ago. -- Odysseus as king is the law. 

Keep in mind that none of Odysseus' men return from the war: all either died in battle or are lost on the return home. The only full citizen males left in Ithaca are those who were too young to go to war (the suitors, Telemachus) or too old to go to war.  In other words, essentially all the men in town are either under the age of 25 or over the age of 60.

We've also already established that, unlike the Hebrews, the Greeks generally do not view their laws and civil morality as divine;  instead, for the Ancient Greeks law is maintained in many ways through brute force: violence. So, imagine that all the police went off to war and never returned, along with the President.  (Note: it's worth point out that at its root democracy itself, which we largely inherit from the Greeks, is predicated on the concept that "might makes right";  that is, democracy is at heart the rule of the majority opinion over the minority opinion and thus at heart the exercise of brute power rather than moral imperative.)