When Woolf wrote A Room of One's Own in the 1920s, many people considered women less intelligent than men. Asserting that rights and resources would be wasted on women because they were inferior was a way to oppose the movement for women's equality. If women were equal to men, people asked, why had none of them produced great literature like Shakespeare's?
Woolf responded that this lack of accomplishment had nothing to with innate ability and everything to do with women's lack of opportunity. To illustrate this, Woolf imagined a Judith Shakespeare, William's sister, heading to London to make her fortune as a playwright. Unfortunately, her male peers view her primarily as a sex object, so rather than being taken seriously as a writer, she is lied to, seduced and impregnated, leading her to commit suicide.
Through this story, Woolf hopes to show that women's lack of accomplishment is cultural, not innate. The culture puts up huge barriers to women's success. A prime one is lack of privacy, leading to the title of the work, but so are the kinds of stereotypes about women that Judith Shakespeare encountered.
Since Woolf's era, scholars have rediscovered a robust literature written by women, and since that time, many women writers have come to the forefront in literature.