Jean Paul Sartre's fiction show examples of characters in such a state.
Some people may never experience it, others only fleetingly, and yet others may live most of their lives in such a state. In extreme cases, a person may enter a 'fugue state' (q.v.) where normal interactions with other people become almost non-existent.
Leibniz's assertion that we 'live in the best of all possible worlds' as satirised by Voltaire's creation of Dr. Pangloss in Candide is possibly relevant here. A person experiencing existential despair generally also believes that they live in the best of all possible worlds - that is, that their situation cannot be improved, and are depressed because their situation feels so bad to them, no matter what an objective, impartial observer may see.
Existential despair is a subjective state, experienced by an individual. If a person so wishes it, it may be amenable to amelioration by drug therapies or 'talking cures'. At least one school of psychiatric therapy, Existentialist Therapy, recognises this state and uses the philosophy of existentialism to help to provide insight into a person's condition.