The opposing senses of the interjection derive from the phrase functioning as an ellipsis of both have no fear ("be assured") and there is no fear of that happening ("it's impossible").
Meanings
Not a chance, impossible.
No fear, therefore, that, suddenly deprived of companionship, he should feel dull, and that dulness might take the shape of remorse; so repent, return, and be forgiven. 1834, L[etitia] E[lizabeth] L[andon], chapter XII, in Francesca Carrara. […], volume II, London: Richard Bentley, […], (successor to Henry Colburn), →OCLC, page 139:
The opposing senses of the interjection derive from the phrase functioning as an ellipsis of both have no fear ("be assured") and there is no fear of that happening ("it's impossible").
Meanings
Synonym of never fear: no worries, don't worry.
"No fear," he said. "Every man will know his own when I come to the Council." 1918, Mary Austin, The Trail Book:
You want me to handle it? No fear.
(UK, dated) Synonym of no way: absolutely not.
"Why, did you show him up some rot?"
"No fear," he said. 1904, M. R. James, Ghost Stories of an Antiquary:
Toad: You should take up swimming!
Rabbit: No fear, toad! 1992, The Animals of Farthing Wood, episode 2:
You're afraid I would handle it? No fear.
Q: Will thou be baptized in this faith?
A: No fear! 1947, Dorothy L. Sayers, Creed or Chaos: