The question deserves more than a surface answer.
Fear of failure — go one level deeper. What does failure mean? Loss of status. Go deeper. What does loss of status mean? Exclusion. Go deeper. What does exclusion mean? Alone. Go deeper. What does alone mean?
The chain always ends at the same place. The Katha Upanishad identifies it precisely: Dvitiyad vai bhayam bhavati — from duality, verily, fear arises. Fear is the specific experience of a self that perceives itself as separate from the universe — and therefore potentially in conflict with it, threatened by it, lost in it.
Every specific fear is a branch of this root. Fear of death is the fear that the separate self will cease. Fear of failure is the fear that the separate self will be exposed as inadequate. Fear of irrelevance is the fear that the separate self will be forgotten.
The Stoic practice of Premeditatio Malorum — the premeditation of adversity — approaches this from a different angle. Seneca's instruction: take the thing you are most afraid of. Sit with it. Describe it in detail. Ask: if this happened, could I survive it? Could I maintain the quality of the inner life even in the middle of the worst version of this? Almost always — with sufficient honest inquiry — the answer is yes. And in that yes, a portion of the fear's power dissolves.
The most useful version of this inquiry is not the philosophical one. Sit with the fear — not to analyse it but to locate it in the body. Notice that what is aware of the fear is not afraid. Something is watching the fear arise. That something does not share the fear's urgency.
That gap — between the fear and the awareness of the fear — is the beginning of the Upanishadic answer. Not the dissolution of fear. The discovery of what is not afraid.