She was born in Kensington, London, one of three children of a barrister, Arthur Frederick Andrew Cole (1883—1968), and his wife, Margaret Henrietta, born Gaselee (1882—1971). She attended St Paul's Girls' School, where she was taught music by Gustav Holst and became the school's organist. She went on to study English literature at Newnham College, Cambridge, and at first pursued a career as a novelist without success. Without giving up this ambition she studied medicine, qualifying as a doctor in 1944. From 1950 to 1963 she worked as a Senior Medical Officer in the Department of Psychological Medicine at the Middlesex Hospital. Afterwards, while regularly producing new children's books, she also worked as an editorial assistant for Penguin Books, from 1966 to the early seventies.

She had met the psychiatrist and author Anthony Storr (1920-2001) during her training and married him in 1942. She had three daughters by this marriage, Sophia, Polly and Emma. They divorced in 1970 and she subsequently married the economist Lord Balogh (1905-1985).

She continued writing novels into her eighties, but became depressed by rejections. She took her own life at her London flat in January, 2001.

Unusually among the leading children's writers of her time, much of her work was for younger children, at the start of their reading, notably the series of stories about Polly and the wolf, which were written for her daughter, Polly. The stories, starting with the collection Clever Polly and the Stupid Wolf (1955), feature a wolf trying to catch a little girl: the wolf, himself a fairy tale figure, takes his always impractical subterfuges from fairy tales, but is outmatched by Polly every time. A novel for slightly older children Marianne Dreams (1958) is more disturbing: a young girl, being tutored at home during sickness, travels in dreams to the house she has drawn while awake and meets there another pupil of her tutor; in a moment of jealousy she draws stones with eyes around the house to keep him prisoner and must then undo her actions. It was made into the TV series Escape Into Night and the film Paperhouse; Storr was not fond of the latter, and particularly disliked the ending.

Storr's books often involve confronting fears, even in the lighthearted Polly stories, and she was aware that she wrote frightening stories. On the subject, she writes: "We should show them that evil is something they already know about or half know. It's not something right outside themselves and this immediately puts it, not only into their comprehension, but it also gives them a degree of power".