In Shakespeare's time, gender roles placed far more limitations on women than they do in the Western world today. A woman's life was to be spent as a wife and mother, and even in childhood girls would be expected to help care for their younger siblings and perform household duties. Women who did not follow the plan of marrying young and having lots of babies until they died were considered to be strange and immoral. Women who did not marry at all were sometimes accused of being witches or sex workers-- which they might well have been, in the absence of a husband to pay the rent!

It is important to remember that in Shakespeare's time, performing arts were considered uncouth. To have to "dance for your dinner" was undesirable and typically only done by people of a low class. Women in Shakespeare's time did not act on stage because it would require them to violate a number of social norms. First, to act on stage, a woman would either have to be unmarried or leave her family at home- neither was deemed acceptable. Second, a woman who performed for money would blur the line between sex work and performative arts. To bring a woman on stage would have been social implication that she was available for sex and could be propositioned openly. Even though they came from lower classes, actors wanted their trade to be seen as reputable and thus distanced themselves from work they considered immoral. 

As women were barred from the stage, women's roles were played by younger men and boys whose voices had not yet deepened. Today, some theatre companies or particular runs of plays which want to stay true to the Shakespearean tradition will cast men in women's roles.

In Elizabethan England, women did not have any personal freedom.  They had no social status outside of their association with their husbands or fathers.  As far as acting, women were normally not educated and did not take on professional jobs.  Acting was entertainment and at times violent.  Therefore it was no place for women. It would be considered immoral for women to be on the stage.  Improper, socially unacceptable.  Associating with men who were not members of your family, without a proper chaperone would be considered scandalous.

One of the reasons that history suggests Elizabeth I did not marry was because she would have lost some of her authority to her husband, it was the way society was organized.