One reason that makes Pride and Prejudice a widely popular timeless classic is that, along with her other works, it paints a very accurate account of human nature. It is completely within human nature to overlook that one's pride has gone too far and that you are acting with conceit. It is also completely within human nature to be prejudiced based on first impressions. It is also within human nature to feel a preference towards one personality type over another and be deceived by that preference, as we see Elizabeth be deceived by Wickham. Furthermore, as is the case with Mr. Bennet, it is completely within human nature to fail at controlling one's family and to fail at teaching them principles.
A second reason that Pride and Prejudice is still so widely popular is that, even though Austen only moralized about principles and society related to her own sphere, those principles and morals are still applicable today.
A final reason is that everyone adores Austen's use of language. She masterfully crafts intelligent irony and wit in a subtle manner that no one, especially no other Romantic author, can compare with.