He was born in Shrewton, a village 4 miles (6.4 km) west of Stonehenge, the eldest son of Alfred and Mary Chubb. Alfred "Fred" was the village saddler and harness maker, as was his father before him. He attended the local village school and then Bishop Wordsworth's School in Salisbury, where from the age of 14 he worked for a time as a student teacher. He met his future wife at a cricket game between his Bishop Wordsworth school and Old Manor Hospital. He then attended Christ's College, Cambridge where he was awarded a double first in Science and Law, leaving with Master of Arts and Bachelor of Law degrees.
In 1902 he married Mary Bella Alice Finch, whose uncle, Dr. W Corbin Finch, owned Fisherton House, which was used as a mental asylum (later the Old Manor Hospital, now Fountain Way). Five years after her uncle's death in 1905, the business and buildings were transferred to her. Following financial difficulties, a limited company was formed to run the hospital in 1924, and Sir Cecil became chairman. Whilst he was in charge, the hospital became the largest private mental hospital in Europe. There is a plaque in the hospital commemorating his life and work. Sir Cecil also served on Salisbury City Council, was a Justice of the Peace and became a successful racehorse owner and breeder of Shorthorn cattle.
Stonehenge was put up for auction in 1915 by the Antrobus family following the death in World War I of the only surviving male heir. Cecil Chubb's interest in the local area led to him attending the sale, with him bidding and purchasing Lot 15 on a whim for £6,600 (about £392 thousand / €498 thousand / $645 thousand today), as he wished to avoid the stones being acquired by someone overseas. It is also speculated that he bought the stones as a present for his wife, only for her to be less than pleased with his new purchase. It is thought that the auction papers remain with Cecil's remaining family in Wiltshire.
Sir Cecil lived at Bemerton Lodge, where Bertie, the future King George VI, was a regular guest. He liked the estate because it was small, secluded and away from London.
Chubb died of heart disease in London on 22 September 1934 aged 58, leaving behind his wife and only daughter.