A hair’s breadth is a very short distance. Breadth in this phrase is synonymous with width, so the phrase literally refers to the width of a hair. Hare’s breath and hair’s breath are interesting images, but for the phrase denoting a short distance, they are misspellings.
On the following Tuesday the world’s financial system came to within one hair’s breadth of extinction. [DeRosaWorld]
Little by little, Mr. Calderón stretches the familiar into a scary, empty new world that is just a hair’s breadth away from the one we know today. [New York Times]
They were a hair’s breadth from mentioning our beloved up-and-coming MFA program. [Rutgers-Newark MFA: Blog]