expression
by a hair's breadth; barely hanging on; still having a tiny chance
Idiomatic expression meaning that one is barely surviving or has only a very slim chance left, often used with verbs like 残る or つながる.
首の皮一枚で試験に合格した。
I passed the exam by a hair's breadth.
会社は首の皮一枚で倒産を免れた。
The company barely avoided bankruptcy by a thread.
Literally 'one layer of skin on the neck,' evoking the image of a neck barely attached by a thin piece of skin, i.e., a very precarious situation.