literally means by the width of a hair. Often when something very nearly happened, or very nearly did not happen we can use this expression to emphasis how close something came. 'By a hair's breadth' is also commonly expressed as 'by a whisker' or 'by the skin of your teeth'.

The "breadth" of something is "how broad" it is. How long -> the length; how wide -> the width; how broad -> the breadth. So as Robin says, the width of a hair! I have seen native English speakers also say/write it as "a hair's breath"... not correct (or logical), but it seems even our own idioms confuse us. ;)

It means by a very narrow margin.