Barr was born in Belmont, Ontario, and was educated at the University of Western Ontario, gaining his BA in 1930, MD in 1933, and MSc in 1938. His association with Western Ontario was continued with his appointment as an instructor in 1936. He subsequently became professor of microscopical anatomy (1952), professor of anatomy and head of the anatomy department (1964), and emeritus professor (1979).

Barr is best known for his discovery, made in 1949 in conjunction with Ewart Bertram, of the densely staining nuclear bodies present in the somatic cells of female humans and other female mammals. These are called sex chromatin or Barr bodies. Later studies by Barr and others revealed that the single Barr body in normal cells is one of the two X-chromosomes in a highly condensed and genetically inactive state. The other X-chromosome is in the diffuse state and is genetically active.

Their discovery enabled Barr and his co-workers to devise a relatively simple diagnostic test for certain genetic abnormalities, in which cells rubbed from the lining of the mouth cavity (a buccal smear) were stained and examined microscopically. For instance, individuals suffering from Turner's syndrome, which usually affects females, have only one X-chromosome and lack Barr bodies. In contrast, males affected by Klinefelter's syndrome possess an extra X-chromosome and exhibit Barr bodies in their cells.

Besides his work in cytogenetics and inherited human disorders, Barr is also noted for his descriptions of nervous-system anatomy. His publications include The Human Nervous System: an Anatomical Viewpoint (1972; 5th edn. (with J. A. Kiernan) 1988).