We know what hair means, don’t we?
let one's hair down
If you let your hair down, you’re totally relaxed or you behave informally.
▲ He finally let his hair down and actually made a joke.
hair of the dog
An alcoholic drink, intended to cure a hangover.
▲ Even a hair of the dog didn't help his aching head, he really shouldn’t have drunk that much wine last night.
A very small amount, degree, measure, etc.; a fraction, as of time or space.
▲ He lost the race by a hair.