First, let me note that Flannery O'Connor first published "A Good Man is Hard to Find" in 1955--about two decades following the end of the Great Depression. Although the setting's date is uncertain, I have always assumed it to be a modern (1950s) story set in about the same period as the story was written. The fact that the family was heading to Florida to vacation--a hot-spot following World War II--would also suggest it to be set in the early 1950s. Truthfully, I see few influences of the Great Depression in the story. The bar-b-que restaurant could well be a remnant from the 1930s, and the dirt road on which the family detoured could also have been unchanged since the Depression. The grandmother's view of African-Americans is certainly an Old South outlook, with no suggestion of being altered by the growing civil rights movements that were beginning to be felt throughout the nation; her view of Negroes is probably unchanged since the Depression era.
To answer the other part of your question, the Great Depression was a devastating economic collapse that affected the world between 1929 until the beginning of World War II. It began with the U. S. stock market crash of 1929, and it plunged the world into a series of economic hardships that reduced personal and business worth by up to nearly 70%. Unemployment in the U. S. rose to 25%; worldwide joblessness was even higher. The Depression affected nearly every sector of the economic strata--rich, poor, and middle-class.
The Great Depression was a severe worldwide economic depression that took place right before World War II. In most countries, it started about 1929 and lasted until the late 1930s and early 1940s. It was the longest, most widespread depression of the 20th century. In the U.S., the Great Depression began with the stock market crash on October 29, 1929. It soon spread worldwide.
In the Flannery O’Connor short story, there is evidence of the effects of the Great Depression. The grandmother was old enough to remember The Great Depression. On the road trip, the family stops at Red Sammy’s Barbecue. This is a beat-up sort of restaurant. The owner, Red Sammy, and the grandmother have a discussion about “hard times” while the family is eating. They reminisce that back in the day, things were different, you could trust people, people depended on each other because times were tough:
The old lady said that in her opinion Europe was entirely to blame for the way things were now. She said the way Europe acted you would think we were made of money and Red Sam said it was no use talking about it, she was exactly right.
The Great Depression was a time of great, long- term economic hardship affecting many nations of the world. In the United States it began with what would later be known as Black Tuesday, (October 29, 1929), this was when the stock market crashed. This resulted in many bank failures and in a dramatic rise in Unemployment. The song "Brother can you Spare a Dime?" was a theme song of sorts for this period in American history as it showed how veterans of the Great War, (World War 1), were now reduced to begging on the streets or waiting in the long bread lines that characterized the time. The Great Depression lasted into the late 1930s and early 1940s.