We all know what a "burden" is, but it's difficult to see how this relates to Jack until all the stories have unfolded. In retrospect, it becomes clear that Jack carries the burden of the secrets of his family, and the secrets of history. He is a living reminder to the Judge, Mrs. Murrell, and Ellis Burden of their secret shames. By finding the truth and bringing it to light, Jack removes the burden, from himself, and from those who have been burdened by it. In the case of the Judge, the results are tragic. The burden of the secret is too great to bear. In spite of all the tragedy, the characters who do live are better off without the burden of the secrets.
Jack also bears the burden of Cass Mastern and his secrets. Those secrets written in Cass's accounting journals that draw Jack in like a magnet. Jack tries in vain to "walk out" on that burden. In the end he that realizes bringing historically significant truth to light is not a burden, but a privilege. This is evidenced by Jack's plan to write a book about what he has learned.
Jack thinks the clues to the characters around him, at least the male ones, lie in their occupations. He even replaces people's names with titles that refer to their occupations. Let's see, we have the Judge, the Governor (this can refer to several people), the Attorney General, the Scholarly Attorney, and the Young Executive. When people ask, he tells them he has no occupation. That's because he will always consider himself a "student of history." Being a student of history isn't a job, it's a talent, a calling. He knows also that Willie's oratory skills (as brought out by Jack's liquor) are a talent. He knows full well that Willie is called the Boss. By the end of the story Jack realizes that occupations only speak to the surface of the characters, while talents (like his for history) are defining. He also learns that talent can be both a gift and a curse.