Hairsbreadth

The speeding car missed the child, who ran out into the road, by only a hairsbreadth.

[...] I ſee you are obſequious in your loue, and I profeſſe requitall to a haires bredth, [...]

[I]t was characteristic of this man, as it is of all men similarly marked, that whatever he did, he overdid, though it were sometimes by only a hair's-breadth [...]

We've missed our shot by a hairsbreadth and that's the same as missing by miles.

In Avatar the spectacular rides to the rescue and hairsbreadth escapes have the whole film behind them and because of that are dramatically and emotionally satisfying.