Eyston was educated at Stonyhurst College and Trinity College, Cambridge, where he studied engineering. His studies were interrupted by World War I, during which he was commissioned in the Dorset Regiment and later served in the Royal Field Artillery. He was awarded the Military Cross on 18 July 1917 for conspicuous gallantry and devotion to duty, having repeatedly carried out reconnaissance under heavy shell and machine gun fire. After the war he returned to Cambridge and served as captain of the First Trinity Boat Club.
Eyston began racing before World War I as a schoolboy, initially competing on motorcycles under an assumed name. After the war he reverted to his own name and moved into car racing, competing in European road races — particularly in Bugattis — with results that included entries at the 1921 and 1926 French Grand Prix.
He became well known in the late 1920s and early 1930s for his exploits with supercharged MG cars, including the Magic Midget and the K3 Magnette. His entries with the K3 included the 1933 Isle of Man Tourist Trophy, the 1934 Northern Ireland Tourist Trophy, and the 1934 Mille Miglia. He also fitted an AEC bus diesel engine into a Chrysler-chassised car to set high-speed endurance records at Brooklands, reaching 100.75 mph in 1933 and 106 mph in 1936.
In 1935 Eyston was among the first British racers to bring a car to the Bonneville Salt Flats in Utah, where he set 24- and 48-hour records with his car Speed of the Wind. He was awarded the Segrave Trophy that year for his achievements.
His most celebrated chapter came with Thunderbolt, a twin-engined car he helped design at the Bean Cars factory in Tipton, Staffordshire. On 19 November 1937 Thunderbolt set a new land speed record at 311.42 mph (501.18 km/h) on the Bonneville Salt Flats, wresting the record from Malcolm Campbell's Blue Bird. Thunderbolt returned the following year with improved aerodynamics and raised the record to 345.50 mph (556.03 km/h) on 27 August 1938. Eyston's fastest run was 357.5 mph, achieved in 1938. He held and lost the record in a friendly rivalry with John Cobb, who twice bettered his marks.
In later years Eyston, serving as competitions manager for Castrol, assisted Cobb in preparations for his ill-fated water speed record attempt on Loch Ness in Crusader in 1952.
As an engineer, Eyston held several patents in motor engineering, particularly relating to supercharging. His invention of the Powerplus supercharger was adopted by MG for their racing cars and contributed materially to their competition successes in the early 1930s. His work on developing high-power gearboxes was also significant in making Thunderbolt's twin-engine configuration viable.
During World War II Eyston served on various industrial bodies and was a Regional Controller for the Ministry of Production. He was made an OBE in 1948. He was also made a chevalier of the Légion d'honneur in 1938, in recognition of his record-breaking achievements.
Eyston authored several books on his racing and record-breaking career, including Flat Out (1933), Motor Racing and Record Breaking (1935, with Barré Lyndon), Speed on Salt (1936, with W.F. Bradley), and Fastest on Earth (1939). He died on 11 June 1979, aged 81.