What is the epiphany in James Joyce's story "The Boarding House" in Dubliners?
Polly's and Mr. Doran's epiphany is that they find themselves in a situation they want to escape from but they can't. Some refer to this feeling of being stuck as a kind of paralysis. The epiphany is the realization of wanting to escape from a certain situation but also realizing that social conventions will make that escape difficult or impossible.
The reason they can't escape from what we are to assume is their inevitable marriage is because of societal pressures. Those pressures are enforced by Mrs. Mooney, Mr. Doran's priest (whom he confesses to), Jack (Polly's brother), and Dublin society in general.
What makes this epiphany perhaps more difficult to swallow (for Polly and Mr. Doran) is the contradictions of adult life in the story. In other words, no one is completely virtuous here. These are never overtly discussed but they are alluded to. Mrs. Mooney feels that Mr. Doran should marry Polly as reparations for essentially taking advantage of her youth. And although Mr. Doran is older than Polly by more than a decade and therefore the more guilty party, it seems that Mrs. Mooney entrapped Mr. Doran to marry her daughter. Remember that Mrs. Mooney runs a boarding house in which the young men refer to her as "The Madam," referring to a woman who runs a whorehouse. The indication is that Mrs. Mooney "knew when to give credit, when to be stern, and when to let things pass" with those young men. One of the benefits of the boarding house (in Mrs. Mooney's perspective) was that it provided her with boarders, young men, and she had to marry off her daughter.
As Polly was very lively, the intention was to give her the run of the young men. Besides, young men like to feel that there is a young woman not very far away.
The indication is that Mrs. Mooney also used Polly to attract business in addition to using the boarding house as a pool of possible suitors.
Other that the epiphanies of Polly and Mr. Doran, the reader might also come to his/her own epiphany. Polly and Mr. Doran are pressured into marriage for societal reasons but those reasons are not necessarily based on ethics. Rather, they are based on economics and conventions. A reader's epiphany might be that Mr. Doran took advantage of the girl but it seems he was simply the one who got caught.