There is a reference to Jane Austen using a circulating library in the Jane Info Pages linked here. See the fourth paragraph.

Subscription libraries were money-making concerns. There was usually a joining fee so that only the “best” families subscribed. They also had a reputation for being full of novels, and therefore not very highbrow. It might be worth mentioning that there didn’t exist anything like public library at the time. There were private libraries and these were mainly scholarly and moral in nature and tended to cater to the interests of men (farming reports, political pamphlets, etc. etc.) but I don’t have more info about them right now.

But in the context of a circulating library, being a “subscriber” meant something different, as in the case of Fanny Price at Portsmouth: “wealth is luxurious and daring” (the residue of £10 ironically called “wealth”!) “And some of hers found its way to a circulating library. She became a subscriber, a renter, a chooser of books”. Here “subscribing” to a library meant paying a fixed fee (often yearly) which gave you the right to take home books for a period of time (of course, the “circulating libraries” were private businesses rather than public amenities…). When books that were expected to appeal to circulating library readers were first published, they were priced fairly expensively and published in a multi-volume format that allowed circulating-library subscribers to be reading different stages of a book simultaneously (so that while one reader was reading volume 2, another would be starting the book with volume 1). In Charles Dickens’ and Charlotte Bronte’s period, about the time that a multi-volume circulating-library edition had pretty much worn out, and the circulating libraries had gotten about as much revenue as they were going to get from the book, the publishers would then print a “cheap edition” priced with individual purchasers in mind, and published with smaller type, in fewer volumes, and made in a less rugged way that was not intended to stand up to normal circulating library wear and tear. (I’m not sure whether this system was quite as fully developed in Jane Austen’s time; certainly no “cheap editions” of any of her books were published in her lifetime.)

In a JA letter of December 18th 1798 (referencing circulating library), she doesn’t refer to Mrs. Martin as a “lady”, and in fact keeping a circulating library was not quite lady-like (for one thing, a shop girl or female shop-owner has to socially interact with, and accept money from, any random male who chooses to come in the door of the shop — something that wasn’t considered fully compatible with being a “genteel” female); however keeping a circulating-library was perfectly respectable, and by no means as radically and totally un-genteel as some other occupations.