One way to approach this assignment is to consider how male characters think and speak about female characters (human and/or divine). How do the males treat female characters? How do the female characters treat the males? What patterns emerge? Based on those patterns, can you identify models of female and male behavior in the Odyssey? In other words, are particular behaviors, conditions, or roles associated with men but not women and vice versa? Do any of the female and male characters violate rules that apply to women's and men's behavior and, in so doing, offer the audience negative models of gender? Reflect on the ending. What do the ultimate fates of the female and male characters reveal about the ancient Greeks' ideas about gender?

A few distinct moments from The Odyssey stick out to me in regard to this topic.  First, early on, Zeus sends Hermes to tell Calypso that she must give up Odysseus and permit him to leave her island to make his way home to Ithaca.  She points out a double standard that really bothers her: gods are allowed to sleep with whichever mortal women they choose, but goddesses are prevented from having relationships with mortal men.  Calypso tells Hermes, 

Hard are you gods and envious beyond all to grudge that goddesses should mate with men and take without disguise mortals for lovers.  Thus when rosy-fingered Dawn chose Orion for her lover, you gods that live at east soon so begrudged him that at Ortygia chaste Artemis from her golden throne attacked and slew him with her gentle arrows.  Again when fair-haired Demeter, yielding to her heart, met Jason in the thrice-ploughed field, not long was Zeus unmindful, but hurled a gleaming bolt and laid him low.  So again now you gods grudge me the mortal tarrying here.

She provides two examples of goddesses who watched their mortal lovers be killed by the gods; they were denied the opportunity to maintain a relationship with mortal men.  She knows that she cannot go against Zeus, but she voices her anger about the fact that female goddesses are not permitted to engage in the same behavior that male gods are.  
Further, when Odysseus returns to his home in Ithaca, he and his son brutally slay the suitors who had been exploiting the rules of hospitality in his home.  One could argue that his actions are justified as a result of their behavior. However, he also has the maids of his home who "consorted" with these suitors hanged.  There seems to be little understanding on his part that these powerless, servant women would have had no real choice if confronted by powerful men.  They would have to be polite, subservient, and even sexually submissive.  Rather than keep this reality in mind, Odysseus kills them too.  Likewise, despite the fact that he has not been faithful to his wife, Penelope, Odysseus expects her to have been faithful to him during his twenty-year absence.
We can see, from these few examples, that there are certainly a number of double standards surrounding gender and sexuality.  Men, including gods, can sleep with whomever they choose, but there are different restrictions on women, including goddesses.  Men are not expected to remain sexually faithful to their wives, though they expect their wives to be faithful.  Worst of all is the lack of recognition that such double standards exist at all and that women are held to different standards than men.