Women in ancient Greece were not considered to be equal to men. In Athens, for instance, women and slaves could not vote - much like the situation in early American history for women and slaves. Women were expected to remain at home, handle the children and the household, and stay out of the affairs of men. This is what Odysseus's wife Peneiope did while her husband went off to fight the Trojan war. After Odysseus was assumed to be dead, Penelope was beset with suitors. She was desirable because she came with all of Odysseus' property. She, however, representing the faithful wife, held off her suitors by telling them she had to finish a tapestry. Every night she would unweave what she had finished, because she felt in her heart that no one could replace her husband, and she did not want to remarry.
The other women in The Odyssey were goddesses or possessed of some supernatural qualities, with the exception of the clever princess Nausicaa, who aided Odysseus in his time of need. Indeed, the mortal women of the story were clever and capable, but, as women, they were peripheral to the more important ideas of honor and glory in battle. The only woman who was a warrior was the goddess Athena, and she was different from other women, since she had sprung from Zeus's head already fully-formed and dressed for battle. She was the patron of soldiers, perhaps representing the anima, or female spirit, as a stand-in for real women.
Even the other goddesses in the story were pictured as petty, vain, and seductive, trying to trap men with their scheming. The Greek society was andro-centric, meaning that it revolved around men. Women were necessary for reproduction and little else. Still, Penelope and Nausicaa were portrayed as sympathetic characters, perhaps the ideal for Greek women to aspire to be.