The story takes place during World War II in Germany. The war began in 1939 and ended in 1945, so the dates of the story take place between these times. The family moves to Auschwitz (what Bruno refers to as “Out-with” in 1942. This would have been the middle of the war, when the concentration camps were full and the Nazi Jewish extermination plan well under way.
The young boy Bruno is clearly in the dark about what his father does. When Hitler visits his house, he seems unaware of his importance. His mother explains that his father needs to move the entire family to Auschwitz.
“…sometimes when someone is very important, …the man who employs him asks him to go somewhere else because there’s a very special job that needs doing there…” (p. 4)
The family does not tell Bruno what he will be doing there, or what the concentration camp is for. When Bruno befriends the little inmates in “striped pajamas” and his friend Shmuel, he ends up losing his life.
I am going to grasp at a few other dates found by scholars to try and get a bit more specific for you in regards to the actual date of The Boy in the Striped Pajamas. If it is true that Bruno was born on April 15, 1934, and it is also true that he is nine years old, then we can be fairly sure the book begins in either 1943 or 1944 (as Bruno would not have turned one until 1935, remember). Remember Bruno's particular pronunciation of "The Fuhrer" as "The Fury," and "Auschwitz" as "Out-With." This would definitely fit with a nine year old with a few slight speech issues.
The years of 1943 or 1944 would also make sense if the reader notes the packed concentration camps and the Jewish people already being exterminated by Hitler's forces. That time period would also make sense in regards to the years of World War II, which lasted from 1939-1945. Because the book is set during World War II, that is the larger span of time during which the story could be set.
This being said, it is very important to realize that, although quite a good example of historical realism, this is a fictional story. In fact, this story has gotten gravely criticized (most specifically from the Jewish community) by glossing over the real truth. The biggest "mistake" that they mention is that there would never be a nine year old boy at a concentration camp, and especially not at the infamous Auschwitz. Those children were not old enough to do any real work, so they were gassed immediately upon entrance.