Mast began racing at sixteen at Natural Bridge Speedway and Eastside Speedway, trading an Angus cow for his first race car. After racing at local track level for a decade, he started running the Busch Series in 1982 and recorded four top-ten finishes in eleven starts. His first full-time Busch season came in 1985, producing fifteen top-ten finishes and a seventh-place points finish. He won his first NASCAR race at the Grand National 200 in 1987 and followed it with another win the following week. He won five additional Busch races across 1989 and 1990 before shifting his primary focus to Cup competition.
Mast made his Cup debut as a substitute for injured Buddy Baker at the Busch 500 in 1988, finishing 28th. He ran thirteen Cup races for Mach 1 Racing in 1989 and notably finished sixth in the Daytona 500 in an unsponsored car — a result he later described as his proudest career achievement, noting he might have won had the team been willing to gamble on fuel strategy.
In 1991 Mast joined Richard Jackson's Precision Products Racing in the No. 1 Skoal Classic Oldsmobile. He led fourteen laps in the 1991 Daytona 500 and finished fourth. That year at Talladega he pushed a fuel-short Harry Gant during the final lap of the Winston 500, helping Gant reach the finish — a gesture that violated NASCAR rules but earned him no penalty. In the same season's DieHard 500 at Talladega, Mast was tagged by Buddy Baker entering the tri-oval, flipped over, and slid hundreds of feet past the start-finish line before climbing out unhurt.
His career-best season came in 1994, with ten top-ten finishes and an eighteenth-place standing in points. He finished second at Rockingham Speedway in a side-by-side battle with race winner Dale Earnhardt, and won the pole position for the inaugural Brickyard 400 at Indianapolis Motor Speedway — a race that drew ninety entered cars. He also won the pole at the final race of the 1992 season, the Hooters 500 at Atlanta, which was simultaneously Richard Petty's farewell race, Jeff Gordon's first Cup start, and Alan Kulwicki's championship-clinching day.
Mast joined Butch Mock Motorsports for 1997 in the No. 75 Remington Arms-sponsored Ford but struggled, finishing thirty-second in points and failing to qualify for the Daytona 500. He joined the Cale Yarborough-owned No. 98 team in 1999 without sponsorship before Universal Studios came aboard mid-season. Mast produced two top-tens and completed the entire season without a single mechanical failure to finish — the first driver to accomplish that for the Yarborough organization since Yarborough himself. Universal did not renew, and Mast found himself without a home again.
His final active seasons saw him move through Larry Hedrick Motorsports, A.J. Foyt Racing, Midwest Transit Racing, and Eel River Racing before finishing the 2001 season with Donlavey Racing. In 2002 he began feeling ill and losing weight, eventually learning he had suffered carbon monoxide poisoning, likely accumulated over years of driving cars whose forced-air induction systems drew fumes from inside the vehicle. He officially retired on January 22, 2003.
Mast's forced retirement prompted direct action by NASCAR on driver safety. After he raised the issue with NASCAR president Mike Helton, the sanctioning body accelerated research into forced-air induction systems. By the Coca-Cola 600 in May 2003, NASCAR had approved carbon monoxide filters for air intake systems. NASCAR subsequently phased out leaded racing fuel beginning in the 2007 season, and the fifth-generation Cup car introduced that year relocated the exhaust outlet away from the driver's position, with carbon monoxide cases like Mast's cited as a reason for the change.
After retiring from driving Mast founded RKM EnviroClean, Inc., specializing in environmental clean-up services, underground utilities contracting, and site demolition. He also served as a stunt double for scenes shot at Daytona International Speedway during production of the 1990 film Days of Thunder. He and his son Ricky began the Mast Cast podcast in 2018.