He was born in Bristol, the son of Francis Augustus Barnett, an iron manufacturer. After leaving Wadham College, Oxford, in 1866, he visited the United States. In the following year he was ordained to the curacy of St Mary's, Bryanston Square, and took priest's orders in 1868. He married Henrietta Octavia Rowland, who had been a co-worker with Miss Octavia Hill and was as much a philanthropist as her husband. Together the couple worked hard for the poor of their parish, opening evening schools for adults, providing them with music and entertainment, and serving on the board of guardians and on the managing committees of schools. Barnett discouraged outdoor relief, because it fostered the pauperization of the neighbourhood. At the same time, the conditions of indoor relief were improved, and the various charities were co-ordinated, by co-operation with the Charity Organization Society and the parish board of guardians.
In 1875 Arnold Toynbee paid the first of many visits to Whitechapel, and in 1877, Barnett, who kept in constant touch with Oxford, formed a small committee, over which he presided, to consider the organisation of university extension in London. His chief assistants were Leonard Montefiore, a young Oxford man, and Frederick Rogers, a member of the vellum binders' trade union. The committee received influential support, and in October four courses of lectures, one by Dr SR Gardiner on English history, were given in Whitechapel. The Barnetts were also associated with the building of model dwellings, with the establishment of the children's country holiday fund and the annual loan exhibitions of fine art at the Whitechapel gallery.
In 1884 an article by Barnett in the Nineteenth Century discussed the question of university settlements. This resulted in the formation of the University Settlements Association, and when Toynbee Hall was built shortly afterwards, Barnett became its warden. He was a select preacher at Oxford in 1895-1897, and at Cambridge in 1900; he received a canonry in Bristol cathedral in 1893, but retained his wardenship of Toynbee Hall, while relinquishing the living of St Jude's. In June 1906 he was given a canonry at Westminster, and when in December he resigned the wardenship of Toynbee Hall the position of president was created so that he might retain his connection. Among Canon Barnett's works is Practicable Socialism (1888, 2nd ed. 1894), written in conjunction with his wife.