How did Elizabeth Bennett overcome gender barriers in Pride and Prejudice?

First, i hope this is not for a paper and therefore cheating. But to answer your question vaguely, think of the similarities between Elizabeth Bennet and Jane Austen herself.

1) Both women refuse to marry without affection

2) Both women refuse the many offers of marriages

3) Both had a Mr. Collins like figure (Jane's being a Mr. John Warren)

the first likeness stated is, perhaps, the most important and most relevant answer to your question. Look at the first few sentences of Pride and Prejudice and you will find that the norm of the day is the opposite of Elizabeth's (and therefore, Jane's) philosophy to happiness, love, marriage, etc.

Start there, and you should find your answer, best of luck