The source of the phrase "by the skin of one’s teeth" is the Bible, Job 19:20. Although the precise phrase Job used was "My bone cleaveth to my skin, and to my flesh, and I am escaped with the skin of my teeth" (not "by").

Just what the "skin" of one’s teeth might be is a bit unclear, but it probably refers to the thin porcelain exterior of the tooth, not the gums. Job evidently kept his teeth, but just barely. It is also possible he was saying the margin of his escape was as narrow as the "skin" of a tooth is shallow — the equivalent of a "hair’s breadth."

In any case, Job clearly meant he’d had a very hard time of it, and the phrase has been used ever since to mean a very narrow or arduous escape.