In "The Storyteller", Saki uses indirect characterization, which is the description of characters, not by telling us information about them, but by telling us about the behaviors and expressions that they make. This, in turn, leads the reader to infer the traits of the characters without having direct information about them. Some of the information that we get during indirect characterization includes the thoughts and dreams (or hopes) of a character, the way that the character behaves and reacts, or the things that the character expresses during discourse. Notice how there is no description of the characters, neither physically, nor internally. Saki does not even give his characters names, with the exception of Cyril, who is one of the boys who misbehaved inside the shared carriage box.
The aunt's character is indirectly characterized by the inability to control the children and her overall righteous mannerisms. Through her actions, we can infer that the aunt is quite a stereotypical, flat character. She is stereotypical because she represents the typical society woman (perhaps a spinster?) with no children of her own who, upon being charged with the responsibility of her nieces and nephew, seems to lose control of them, and of her own discipline skills. Since she does not change throughout the story, she remains a flat character.
The bachelor is also flat because he does not experience any significant change as a result of the plot. If anything he does move the plot toward the end when he tells the story of Bertha to the children, and causes them to like the story. From the indirect characterization of the bachelor, we can infer that he is a bachelor precisely because of his dislike of family bundles, such as that of the aunt and the children. He is also creative and willing to do anything to save himself from being annoyed, even if it takes to come up with a story to calm down the rowdy children. Moreover, the bachelor also seems to have an upper-hand when it comes to people skills, for not only did he calm the children, but he also bragged about his superior skills to the aunt.
It is also acceptable to include that the bachelor is also stereotypical of his time and historical context because the character of the "eternal bachelor", that is, the uncommitted, thirty-something year old man who wishes to live his life away from the hassles of family life, is a character-type that repeats itself all throughout Victorian and post-Victorian literature.
The children, although moved by the story, are also flat. No transformation comes their way with the exception of their liking the story which, itself, does not constitute a character's transformation; they will continue to misbehave more than likely once the excitement of the story wears off. Hence, through indirect characterization we also know that they are mischievous children, prone to misbehave, and cannot be controlled by their aunt.
Therefore, through indirect characterization Saki presents to us characters who are rather flat and stereotypical of their historical time in context. The fact that they are flat characters (or static) does not entail that they are bad, incomplete, or not good enough; it simply means that the plot does not alter their behaviors, views on life, or their thoughts.