breadth的意思是“宽度、幅度”，那a hair's breadth字面意思就是“一根头发的宽度”，引申意思是“千钧一发、十分靠近、极小量、间不容发的距离”。

【例句】
Fortunately, he was sent to the hospital in time. He came within a hair's breadth to losing his life.
幸运的是，他及时被送往医院，千钧一发地捡回了一条小命。

She was within a hair's breadth of winning.
她差一点就赢了。