Baker was born on 19 March 1901, the son of J W Baker (of Wallasey) and Emily C Westwood. He was educated at Rossall and Clare College, Cambridge and in 1928 married Fiona Mary MacAlister Walker (d.1979).

Baker held many scientific and academic positions including: Technical Assistant, Design Department, Royal Airship Works, 1925; Assistant Lecturer, University College, Cardiff, 1926; Scientific Assistant, Building Research Station, 1928; Technical Officer to Steel Structures Research Committee, 1931-36; Professor of Civil Engineering, Bristol University, 1933-43; Professor of Mechanical Sciences and Head of Department of Engineering, Cambridge University, 1943-68, Professor Emeritus, 1968; Chairman of the Council, School of Physical Sciences, Cambridge University, 1945-72. He was a member of the Civil Defence Research Committee, 1939-1948; the Scientific Advisory Commission, Ministry of Works, 1945-46; the Advisory Council to Military College of Science, 1947-52; the University Grants Committee, 1953-63; a member of Council, British Welding Research Association; President of the Welding Institute, 1971-73 (Brooker Medal, 1977); President of the British Association for the Advancement of Science, 1975-76; Chairman of the Naval Education Advisory Committee, 1958-64; Consultant, Naval Constructional Research Establishment, Rosyth, 1948-63. He was also a Director of companies including: Technical Development Capital Ltd, 1962-74; John Brown & Co. Ltd, 1963-71; Cambridge Fender and Engineering Co. Ltd, 1964-74. Baker was also a Scientific Adviser and in charge of Design and Development Section of the Ministry of Home Security, Air Raid Precautions Department, 1939-43. In 1940 he designed the Morrison indoor shelter.