Fin (Peter Dinklage)--short for Finbar--is an achondroplastic dwarf and a taciturn lover of trains. He repairs toy trains in a shop run by a tall elderly black man. When the shop owner dies suddenly and bequeaths Fin a "house" in Newfoundland, New Jersey, Fin, jobless, uproots himself to seek out his inheritance. The house turns out to be a deserted, former Station House adjacent to train tracks and is located in an abandoned section of the community.

Fin tries to make the house livable, sleeps on a sofa, and relocates the outside mailbox so that he can reach it. Once he can demonstrate that he receives mail and pays bills at that location, he applies for a library card--he is an avid reader of train lore. Fin seems content to sit on an outdoor bench, clocking the trains that pass by, reading his books, walking the tracks, and keeping to himself in his little house.

Two people interrupt his solitude: Joe (Bobby Cannavale), a gregarious young man who has taken over his sick father's food truck stand--and Olivia (Patricia Clarkson), an artist and divorcee who twice nearly runs Fin off the road in her small SUV. Joe tries repeatedly to engage Fin in conversation and comraderie; Olivia makes fumbling apologetic overtures to Fin. Fin grudgingly begins to engage with Joe and Olivia and they become a threesome as Joe and Olivia follow Fin on his train track walks, sit with him as he clocks trains, and share dinner at Olivia's waterfront home.

Each of the three protagonists is a wounded soul. Fin endures startled glances, snickering whispers, outright rude comments, and even invisibility--a supermarket cashier passes him over for the next customer because she does not see him; he longs for a "normal" body that would allow him to physically defend himself; he longs for a normal sex life. Joe is "happy-go-lucky" on the surface, but is under the thumb of a domineering father who makes frequent calls to Joe's cell phone. Joe tries, unsuccessfully, to court Olivia. Olivia is enveloped in guilt and mourning over her young son's death and thinks she is still in love with her former husband.

Two other individuals play a role in Fin's new life: the pretty, young librarian (Michelle Williams) who tells Fin that he has "a nice chin" and confides to him that she is pregnant by her boyfriend, a boorish local she has not yet told; and Cleo (Raven Goodwin), a preteen black girl who is curious about Fin's train knowledge, and seeks his friendship. Cleo enlists Fin, against his will, to speak about trains to her school class.

Olivia triggers Fin's outburst of pent-up rage and frustration: she rejects his concerned vigil, when, for days on end, she refuses to leave her house or answer her telephone. The despondent Fin goes to the local bar, downs glass after glass of whiskey, sitting alone; thoroughly drunk, he smashes his glass, climbs up on the bar, gesticulating and yelling at the crowd to "go ahead, look at me, here I am!" (paraphrase). Staggering out onto the train tracks, he falls as an approaching train barrels down on him. He smiles up into the train lights, seeming to welcome what appears to be certain death.