Hesketh Racing entered Formula One using a March 731 chassis before commissioning Postlethwaite to design a bespoke car. The 308 was loosely based on the March architecture but incorporated several novel engineering ideas. Power came exclusively from the Ford-Cosworth DFV V8; a planned Hesketh-commissioned V12 engine never materialised.
One of the 308's distinctive technical features was its rubber spring suspension. Postlethwaite was seeking lightweight progressive springing and, following a suggestion from a contact involved in earthquake-damping for buildings, investigated rubber springs. Early tests in 1974 were unsuccessful until a non-creep compound was developed by the Malaysian Rubber Producers Association and formed into springs by Aeon Products. The system was fitted from the 1975 Argentine Grand Prix onwards, noticeably improving the car's balance. Four chassis were built across the three seasons the car competed.
The 308 made its public debut at the 1974 Brands Hatch Race of Champions, where Hunt immediately put it on pole position. The race itself was less successful — Hunt spun out in the wet on lap four. Two weeks later at the South African Grand Prix the car made its World Championship bow; Hunt qualified thirteenth but carved through to fifth place before a driveshaft failure ended his race on lap 65.
At the non-championship International Trophy at Silverstone, the potential of the package was vividly demonstrated. Hunt started from pole, dropped back with a slipping clutch and the gear-shift knob coming off in his hand, yet fought his way back to the front, passing Ronnie Peterson's Lotus through Woodcote corner to take a stunning win.
The Grand Prix season proved tougher. The 308 was quick but fragile, suffering repeated transmission failures. Hunt was also involved in two accidents with Tom Pryce — in the Netherlands, where Pryce collided with him at the first corner, and in France, where Pryce was pushed into Hunt by Carlos Reutemann. Hesketh briefly ran Ian Scheckter in a second car for the Austrian race, though engine trouble prevented him from qualifying.
For 1975 the car was revised to 308B specification with updated bodywork and repositioned oil radiators. The improvement was immediate: Hunt challenged seriously for victory in Argentina and Brazil before finally winning at Zandvoort. The Dutch Grand Prix result stood as the only World Championship win for the team, but consistent points finishes through the remainder of the year lifted Hesketh to fourth in the Constructors' Championship.
The team folded at the end of 1975, unable to continue financing an unsponsored effort. Hunt was signed by McLaren for 1976, with whom he would go on to win the World Championship.
The 308 and 308B chassis were sold on to privateer entrants, who achieved modest results at best. The most prominent subsequent user was Guy Edwards, who arranged Penthouse magazine sponsorship for 1976, generating considerable media coverage, though no points were scored.
The Hesketh 308 occupies an affectionate place in Formula One history as the product of an unsponsored, aristocrat-funded team run with a sense of fun that was at odds with the increasingly professional paddock of the mid-1970s. Harvey Postlethwaite's design showed genuine ingenuity in its rubber spring suspension system, and the car's sole victory at Zandvoort launched the career of a future World Champion. It remains a symbol of the last era in which a well-heeled private entrant could credibly challenge the established constructors.