How did Jem remain calm when Miss Maudie's house was burning and it looked as though the Finch house would burn also?
I agree with the previous post concerning Jem's calmness during the house fire in To Kill a Mockingbird. When Atticus awakes the children, he calmly leads them outside, assures them that the house is probably lost but that all will be well, and then ushers them to the relative safety of the Radley Place. Jem is also busy keeping Scout calm, hugging her and telling her that "It ain't time to worry yet. I'll let you know when." It is one of the few times in the novel where he so outwardly shows his brotherly love for his sister. They both found Atticus' own calm demeanor reassuring.
... Atticus was standing with his hands in his overcoat pockets. He might have been watching a football game.
"See there, he's not worried yet," said Jem.
They also could plainly see that Atticus was not going to risk his life by trying to save furniture or other valuables, so they knew he would be safe. "He's too old, he'd break his neck," Jem assured Scout. Besides, it was such an exciting event that the kids had no time to become scared.
You are talking about what happens in Chapter 8 -- that is where the part with the burning house is. Towards the end of the chapter is where you can find the answer to this.
The answer, I believe, is that Jem remains calm because of the example of his father. He looks over at one point and sees Atticus being calm. When he sees this, he tells Scout that it is not time for them to panic -- they can be calm -- and it is implied that this is because Atticus is calm.