Scout is eight years old when Jem breaks his arm in an altercation with Bob Ewell.

In the first sentence of the story, Scout mentions that Jem was "nearly thirteen" years old when he broke his arm. Later in the first chapter, Scout mentions that she was "almost six" years old when Jem was nearly ten, which means that Scout is four years younger than her brother. Given the fact that Scout is four years younger than Jem, she would have been eight years old when Jem broke his arm.

There are also several moments towards the end of the novel where Jem mentions Scout's age. In chapter 17, Jem tells Reverend Sykes that Scout can't understand the obscene details of the trial because "she ain't nine yet" (Lee, 175). In chapter 23, Aunt Alexandra attempts to explain why Scout cannot play with Walter Cunningham Jr., and Jem once again comments on his sister's age by saying, "she ain’t nine yet" (Lee, 228).

When he was almost thirteen,  my brother Jem got his arm broken badly at the elbow. 

The beginning of the novel opens with this statement from Scout, in which she provides the reader characterization of her brother Jem.  From the beginning, the reader can infer how much Scout looks up to her older brother, who is four years older than her. 

"I maintain that the Ewells started it all, but Jem, who was four years my senior, said it started long before that."

By doing a little simple math, the reader can conclude that when Jem broke his arm at the age of thirteen that Scout would have been nine years old.  As the true action of the story begins, Jem is actually ten, and Scout is six, the consummate tag-along little sister.