A living in the sense you're asking about is not as generic as that referenced in the phrases "doing X for a living" or "it's a living" - in nineteenth century novels, a living is specifically tenancy as a clergyman in a parish. The right to choose the next tenant was usually owned by one of the local bigwigs, which sometimes meant - as with the Crawleys in Vanity Fair - that the landowner would pass down the money and property to his eldest son and grant the living to the younger, if the second son had gone into the church rather than the law/Parliament or the military. They could also sell the next tenancy to someone else if they had no need of it. (Forgive me if you know this already, but it's not clear from the question.)

A living could be either a rectory or a vicarage: on both cases, the clergyman and his family would get a certain amount of land to rent out or till for personal use, as well as the "lesser tithes", ten percent of the livestock and non-wheat crops produced in the parish or their monetary equivalent, but only a rector would also get the "greater tithes", ten percent of the wheat crops or their monetary equivalent. When the living was a vicarage, the landowner who had the right to appoint the clergyman would get the greater tithes. All of this was very helpful to support the family during the life of the rector or vicar, but since it was tied to the tenancy and could not be passed on, there was a certain precarity to it, which is reflected in fiction. In Mansfield Park, Mrs. Norris is the widow of the last tenant of the living, and has to move in with her sister's family, who own the living in the first place; she's much better off than Mrs. and Miss Bates in Emma, who are left with no form of support at all when Mr. Bates dies. (Jane Austen herself experienced this, as her father was a vicar who died and left his wife and daughters dependant on his sons.) In North and South, Elizabeth Gaskell showed the way even a living clergyman could be left penniless by vacating his living over a matter of principle.

The prerequisite for getting a living was having been educated religiously - getting a university degree in theology. A huge number of Oxford and Cambridge (and Dublin's Trinity) graduates in the eighteenth and early nineteenth centuries went for ordination, to become Anglican ministers. The rising number of young ordinated men with the static number of livings (10,500) made competition fierce! In addition, they had to have testimonials for their moral conduct, preferably from three older clergymen. So while middle-class men were legally able to join the clergy, they would have a hard time affording and getting into university and likely in finding testimonials, unless their fathers were also in the church.