In chapter 3, the narrator relates how handyman Homer Smith drove the nuns...
American author William E. Barrett’s novella The Lilies of the Field was published in 1962. No specific dates are mentioned in the novel, so it's difficult to determine when the action of the story actually occurs.
In chapter 3, the narrator relates how handyman Homer Smith drove the nuns to Mass on a Sunday morning, but "this close was as close as he [Homer] intended to go to a Catholic Mass." So when the Mass began, Homer crossed the street to a small cafe where he ordered breakfast—"Ham and eggs, with lots of ham, and pancakes, and anything else you’ve got, and coffee"—and talked with the man behind the counter, who was also the short order cook who prepared his breakfast.
As the narrator says, the "little man" behind the counter seemed to know all about the nuns. "They were from the wrong Germany, 'the one that is Communist,' and they escaped" because they couldn't stay in Germany (chapter 3).
The Berlin Wall, which divided Communist East Berlin and the rest of East Germany from West Berlin and West Germany, was erected in 1961. It seems reasonable to assume that the nuns were forced out of East Germany at some time in 1961 (construction of the wall started in August 1961) or early 1962 when the city of Berlin and the rest of Germany were divided.
The nuns then had to make their way from Germany to Arizona, where the nuns had been left a plot of land in the will of "a hard, mean man" named Gus Ritter, whose sister had belonged to the same order of nuns.
In the meantime, Homer Smith had been discharged from the army from where he had been stationed at Ft. Lewis, Washington. "He bought a secondhand station wagon in Seattle, equipped it for sleeping, and started out to see the West" (chapter 1).
"On a morning in May, Homer Smith drove into a valley west of the Rocky Mountain Range" where the nuns had their plot of land, and where he started to do odd jobs for them (chapter 1).
Homer stayed in the valley from May until September, doing odd jobs for the nuns and eventually building a "chapel church" for them. After supper one evening in September, Homer decided that it was time to leave the valley.
The starter awakened the engine of the station wagon to noisy life and a touch of the switch sent a flow of light down the rutted road. Homer Smith drove that road to the highway and he knew that he was never coming back. A man couldn’t roam forever, nor pleasure himself in strange cities indefinitely. There was a settling down time at the end of all that. The road was lonely and he sang softly to himself as he drove. (chapter 6)
The narrator mentions that "the legend of Homer Smith came into being within twenty-four hours of his disappearance from the scene of his labors" (chapter 7).