Dr. Eyman was born in Fairfield County, Ohio, and educated at Fairfield Union Academy. He taught school for several years, then entered the Columbus (Ohio) Medical School, graduating in 1880. He began his medical practice in Tarleton, Pickaway County, Ohio, but ill health forced him out of practice, and for a time he operated a drug store with his brother in Lancaster, Ohio.
In 1877, he became assistant superintendent of the Toledo State Hospital. In 1884, he was appointed consultant physician at the Athens State Hospital. In 1891, Dr. Eyman was appointed superintendent of Cleveland State Hospital. He later became superintendent of the Massillon State Hospital in 1903, remaining until 1918.
At the Cleveland State Hospital, Dr. Eyman abolished mechanical restraint for patients. He worked for 12 years as Professor of Nervous and Mental Diseases at the Cleveland College of Physicians and Surgeons. Dr. Eyman was a member of the American Medical Association, an honorary member of the Canton Medical Society, and served as Secretary (1915–18) and President of the American Medico-Psychological Association (1919–20).