Born in Waterford, Barker was one of six children of William Barker (1731–88) and Elizabeth Barker (née Acheson).[1] Barker was educated in Edinburgh during which time he got to know Sir Walter Scott.[1] He graduating as an MD in 1795, with his thesis studying the work of Galvani that suggested the presence of nervous fluid with dynamical electricity.[2]
Barker established what is believed to be the first fever hospital in Ireland in 1810.[2] He was Professor of Chemistry at Trinity College, Dublin (1808–50), and Secretary to the Irish Board of Health (1820–32).[1]
Barker married Emma (née Conolly), the daughter of the vicar of Donard, County Wicklow in 1804.[1] They had four daughters and a son William (1810–73), who went on to be a professor of chemistry (1850–73) at the Royal College of Science for Ireland.[1]