Born in Dublin, Barker attended the School of the Royal College of Surgeons in Ireland and then Bonn University. Initially appointed surgeon at the City of Dublin Hospital in 1876, he then joined the staff of University College Hospital, London, becoming surgeon in 1885. In 1893 he was appointed Professor of the Principles and Practice of Surgery and of Clinical Surgery at the Medical School attached to the College.
As a general surgeon, Barker was a pioneer of abdominal operations, performing the first successful gastroenterostomy for malignant disease. His other areas of expertise are recorded as leucoplakia of the tongue and neurosurgery.
His interests also included spinal anaesthesia where he is believed to have been the first to use glucose along with the local anaesthetic in the injection fluid. This technique was demonstrated by his “glass spine”, a tube bent to represent the space in the spinal column containing the spinal cord and cerebrospinal fluid.
A temporary Lieutenant-Colonel in the Royal Army Medical Corps in Southern Command during World War I, he died while on active service from pneumonia and nephritis.