Born in Australia, Stable spent much of his early life in Switzerland, where he lived with an aunt. He was educated at the College de Geneve (Switzerland), and then Cambridge University, where he graduated with an M.A. in 1909. After spending several years in Germany, in 1912 he took up a position as lecturer in English, French and German at the University of Queensland. He was later appointed Professor of English Language and Literature at the University of Queensland, and on his retirement in 1952, was made Emeritus Professor.
Stable made a significant contribution to Queensland cultural life. He was at various points president of the Queensland Authors' and Artists' Association, and the English and Modern Languages Association of Queensland, and he was also a founding member of the Brisbane Repertory Theatre Society and the Queensland Historical Society, as well as being a trustee of the Queensland National Art Gallery. For a number of years he also contributed articles of literary criticism to the Brisbane Courier.
Stable has been widely recognised for promoting the study of Australian literature, at a time when it attracted little interest within Australian universities.