Both Jane Austen and Elizabeth as the primary character make it quite evident that they view Mr. Bennet's escape to his private library as a foolish escape from both reality and his masculine duties as head of the household.
After Lydia runs away with Wickham, we see Elizabeth reflect on the fact that, while she has always admired her father, especially for his affection, she has always been aware of his deficiencies as the head of the household. Mr. Bennet made a very foolish mistake in marrying Mrs. Bennet solely based on her beauty. Unfortunately, Mrs. Bennet also came from a lower class rank than Mr. Bennet, the merchant class, and has turned out to be a very ridiculous woman. So ridiculous that she even encourages her younger daughters to be dreadful flirts. Hence, Mr. Bennet observes all of the ridiculous, foolish, and improper behavior going on around his household due to his poor marriage decision, but chooses to hide himself from it in his library. As the narrator relays to the reader that he once explained to Elizabeth:
In his library he had been always sure of leisure and tranquility; and though prepared ... to meet with folly and conceit in every other room of the house, he was used to be free from them there. (Ch. 16)
Thus, Elizabeth reflects that instead of hiding in the library, his time and his intelligence would be better used in actually trying to discipline his family, which might have saved Lydia from her indiscretion and spared the Bennet family any disgrace. We see Elizabeth reflect on the fact that Mr. Bennet could be using his intelligence to save his family when she states that she had never "been so fully aware of the evils arising from so ill-judged a direction of talents; talents which rightly used, might at least have preserved the respectability of his daughters" (Vol. 2, Ch. 42).
Hence we see that the significance of Mr. Bennet's masculine sanctuary is that he uses it as an escape from his poor decisions and his responsibilities as head of the household, much to Elizabeth's as well as Austen's disappointment.