There are innumerable reasons why Jane Austen's books have continued to gain popularity since their publication around 200 years ago. Actually, this is the 200th anniversary year of the publication of her first novel, Sense and Sensibility (1811). But I digress. Austen's work is generally lauded for her characteristic gentle sarcasm and mocking narration. She had an amazing understanding of how people think, and developed her characters in a very natural and detailed style. Her stories are well laid-out, and entertaining (which is important in entertainment). Jane (unlike many authors nowadays) wrote about what she knew, and I think that's one of the primary reasons that what she wrote has stood the test of two centuries.
They are all romances, 'tis true. In 19th century England, the only way for a middle-class woman to secure a comfortable life was to marry well. It was the predominating fact of life for women then. From her stories, it's easy to gather that Jane rebelled against the all-to-common method of making the best "match" possible; calculating women plied their feminine wiles, and gambled with their futures trying to get the wealthiest and handsomest man they could, without waiting too long and having to settle for whoever they could finally get. Jane's heroines are above this mentality. As Elizabeth Bennet said in Pride and Prejudice, "I am determined that nothing but the very deepest love could ever induce me into matrimony." That certainly appears to be Jane's own opinion on the subject, as all of her heroines married for love, and those couples who did not marry for love were portrayed as quite miserable, and living with their just deserts. Also worth noting is that the couples who did marry for love, while being "perfect" personality matches for each other, were still not flawless people (an aspect of reality that is sometimes left out of romance stories).
I believe these stories are still so popular, particularly with women, because even though our society no longer requires women to marry to acquire comfortable lives, marriage is still built into the essence of our very beings. Woman was created for man. I believe every woman has a place in her soul that longs and yearns to be loved and belong to that one man for her whole life. Hollywood and cheap authors are cranking out movies and books with romantic stories, but the men are usually shallow, women are objectified and "til death do you part" is completely ignored. If a man "loves" a woman in these stories, it's usually just lust in disguise and that's what makes Jane Austen's stories so much richer in comparison. Yes, there are shallow men who only lust after women, but there are also respectful, honorable men who truly love and pursue the woman they admire. Mr. Darcy, Mr. Knightley, Edward Ferrars, Colonel Brandon, etc, are all different portraits of honorable men. The kind of men that women still long for, and sometimes despair of existing.
It doesn't matter that these stories are set in an era totally different from our own, with lives completely unlike the lives of modern Americans, fundamentally, I believe people will always be the same. There will always be callous, uncaring cads, there will always be silly girls who will fall for them. There will always be more sensible women who want real men. And though he most likely will not dress, act, talk or look like Mr. Darcy, we hope that there are still men who are honorable and noble. We still hope for our Christlike knight. Don't worry about the shining armor or elegant finery of Mr. Darcy. Jeans are fine.