From a Westmorland family, he went to Queen's College, Oxford, where in 1778 he gained the chancellor's English essay prize on academic education.

He preached as the Bampton lecturer for 1799, on Answers to some Popular Objections against the Necessity or the Credibility of the Christian Revelation.

He was indebted to William Paley's writings for the argument; he popularised arguments for the necessity and probability of a divine revelation to man, that the doctrines and precepts of the Christian religion are favourable to the enjoyments of the present life, and, with regard to prayer, deemed it probable that "the Almighty in consequence of our prayers interferes with the laws of nature".

This was not separated at that time from the province of York, and was held by Barrow for two years, until age and infirmity caused him to resign it to Dr. George Wilkins in 1832.

He also published two sermons which had been preached at Southwell before the loyal volunteers of that place during the panic of 1803–4, and another on Pecuniary Contributions for the Diffusion of Religious Knowledge; a treatise on the Expediency of translating our Scriptures into several of the Oriental Languages and the means of rendering those Translations useful (1808), Familiar Dissertations on Theological and Moral Subjects (1819), and three volumes of Familiar Sermons (1818–21).