What does Mark Twain mean in his statement about Austen's novels, relating his statement to Emma: Austen's novels make him feel "like a barkeeper entering the kingdom of heaven"?
This simile comes from a fragment of a critical opinion by Twain on Austen titled "Jane Austen." A fragment is a piece that was begun by an author and, for one reason or another, never completed. This Twain fragment is stored at the University of California-Berkeley. Twain made this remark in this fragment in relation to Pride and Prejudice and Sense and Sensibility but it may be applied to Emma as well because Austen wrote about the same social sphere--her own--in all her novels.
What this Twain simile means, in terms of its specific language, is that when Twain reads Austen, he feels as unrefined, unpolished, uncivilized as would a scroungy Mississippi River barkeeper upon first entering polished, shiny, orderly, well-mannered heaven. They'd both feel out of place, out of their element, overwhelmed by manners and speech they had never encountered before--and--slightly repulsed by it for it's strangeness and newness and for its opposition to everything they'd previously encountered. You might paraphrase Twain's quote like this: What is this place and who are these people and whatever is the matter with them; and what in thunder am I doing here with them!?!
There is some debate as to whether Twain was actually being truthful in his vitriol against Jane Austen or whether he was applying that famous "Jumping Frog of Calaveras County" satirical sarcasm and irony to a social world and elegant, educated writing style he simply could not fathom yet secretly admired since, like the barkeeper, he kept going back and trying again to read Austen's works.
Yet [the barkeeper] would be secretly ashamed of himself, secretly angry with himself that this was so. Why? Because ... there are fine things, great things, admirable things, which others can perceive and [he] can't. [...] So he tries again. (Twain, "Jane Austen")
Let's apply this to Emma. In Emma, which Twain does not address in his fragment, Twain would find the heavenly world filled with degrees of elegance, wealth, power, finesse, and polite, educated respectability in the Churchills, Frank Churchill, Mr. Knightly, and Mr and Emma Woodhouse. These would be the prominent heavenly "Presbyterians," as he called them in the fragment. He would find whist tables, balls, outings in open carriages to Box Hill, strawberry parties, tree lined avenues with sweeping vistas and heroic gallantry that rescues scorned or threatened ladies (Knightly and Frank).
Yet, when Twain looked a little deeper, he found the sordid and surprising (as he acknowledges in character sketches in the fragment). The sordid in Emma would be the "gipsies" and Harriet's mysterious, unknown parentage. There would be acrimony amongst Box Hill party-goers and bigoted disdain toward the unfortunate Harriet at the ball. The surprising would be:
-  the town clergyman having a vendetta against the town patroness: Elton against Emma.
-  children sent away to be raised "out" by wealthy families: Jane Fairfax and Frank Churchill.
-  secret engagements and scurvy flirtations to cover up the secret.
-  gossiping speculations of sinister unrequited love affairs between a married man and a single woman: Mr. Dixon and Jane Fairfax.