A town named Summit, presumably outside of Chicago, sometime during autumn
Before talking about "The Killers," it’s a good idea to think about what was going on at the time Hemingway wrote it. We’re looking at the mid-west in 1927, which means two big things: Prohibition and organized crime in Chicago. They’re related. When alcohol was illegal, the mafia made a lot of money supplying it. Prohibition is definitely in effect at the time "The Killers" takes place; look at the line where the killers pointedly ask George if he has anything to drink. They’re asking for alcohol, but he responds only with non-alcoholic options. In the 1920s, Al Capone was running the mafia show in Chicago, so he would have been in the front of reader’s mind at the story’s mention of this city and of mob activities.
What mob activities? Well, the killers pretty much give the game away when Max says that they’ve never met Ole but are killing him "for a friend." Later on we hear that Ole "must have got mixed up in something in Chicago," so we’re thinking that, as a prizefighter, he didn’t go along with fixing a fight and pissed off some pretty important Mafiosi. Hence the hitmen.
One last thing: vaudeville. You can read in the Character Analysis for Al and Max that the two of them act as a sort of vaudeville duo. Again, this makes a lot more sense if you’re thinking about it from a 1927 mindset instead of a 21st century one. By the 1920s, vaudeville had been popular for around forty years – readers would have known the form well.
Now what about the actual locations where the story takes place – the lunchroom and the boarding house? Take a look at the names. We find out in the first line that the diner in question is called "Henry’s lunchroom." Except George is the guy running the place. Who is Henry? We don’t know. Then you take a look at the boarding house, which is owned by Mrs. Hirsch. But the landlady we meet is Mrs. Bell. Where’s Mrs. Hirsch? We don’t know. This looks like another case of things not being what they seem, and further, of uncertainty. In this way, the setting compliments the themes, which is always nice.