Oh, quite far! Her father was from a long line of wealthy wool merchants who had made it into the gentry over the generations, but her mother, a Leigh, was from an old family with strong aristocratic connections, and grew up in Oxford among gentry, her father being the rector of All Souls (which is how she met George Austen, who was a student at St John's).
Even though both of them were from poorer branches of families with considerable wealth (George Austen and his sisters were orphaned young and taken in by relatives -- his sister Philadelphia, without a dowry, went out to India to marry, did so, and her daughter in turn, who knew JA and Cassandra well (and may have been the model for JA's Lady Susan), married a French count who was guillotined during the French Revolution and later married JA's brother), they still benefited from having rich and well-connected family members helping them.
George Austen was given the living of Steventon by his wealthy second cousin's husband, and a neighbouring living was bought for him by a wealthy uncle. Their income was never big, needing to be supplemented by farming and taking in boys to educate, but they absolutely socialised as gentry during JA's youth, attending assembly balls with people on a level with the Bennets and above, and dining with them.