Born in Olomouc, Moravia, then part of Austria-Hungary, Junková was the sixth of eight children of a locksmith. Her multilingual ability earned her work at the Prague Credit Bank's Olomouc branch after World War I, where she met young banker Vincenc "Čeněk" Junek. The two married, and Eliška adopted the feminine Czech form of his surname, becoming Junková. The couple moved between Brno, Prague, and eventually France, where Čeněk grew wealthy enough to indulge his automotive passions. Eliška initially resisted, but quickly fell for the appeal of the high-performance cars of the era, particularly Bugattis. She obtained her driving licence through clandestine lessons in Prague in 1922.
The Juneks purchased a Mercedes and then a Bugatti Type 30 that had previously been raced in the French Grand Prix. Eliška initially served as riding mechanic to her husband, but a wartime hand injury affecting his gear changes gave her the opportunity to take the wheel. Her first professional race came in 1923, with Čeněk at her side; the following year she raced independently, winning her class at the Lochotín-Třemošná hillclimb and becoming an overnight national celebrity.
She placed first at the Zbraslav-Jíloviště hillclimb in 1925, the same year the couple became close friends with Ettore Bugatti. By 1926 she was competing across Europe, finishing as runner-up in the prestigious Klausenpass hill climb in Switzerland. Her technical approach was innovative: she was among the first drivers to walk a course before an event, noting landmarks and memorising the optimal racing line. She was identifiable on track by her consistent attire of blue skirt, white blouse, helmet, and goggles.
In 1927 she entered the Targa Florio in Sicily for the first time, driving a 2.3-litre Bugatti Type 35B with Čeněk as her mechanic-passenger. After completing the first of five laps in fourth place — only ten seconds behind the works Bugatti of Emilio Materassi — her run ended when the steering gear broke and she was forced off the road. Neither was injured. That same year she won the two-litre sports car class of the German Grand Prix at the new Nürburgring and the 50-kilometre ladies handicap at Montlhéry during the 24 Hours of Paris.
Her most celebrated performance came in the 1928 Targa Florio. Junková arrived in Sicily a full month before race day, completing thirty to forty reconnaissance laps and marking corners with chalk. On race day she started well into the second half of the grid but was fourth on elapsed time by the end of the first lap, barely thirty seconds behind leader Louis Chiron. When Chiron retired and Albert Divo pitted, Junková moved into the overall lead — an unprecedented position for a woman in a top-level Grand Prix race.
Going into the final lap she was still second overall and within a minute of leader Giuseppe Campari. A puncture in the closing stages cost her two vital minutes while Čeněk changed the tyre; the stop overheated the engine and she was forced to nurse the car home. She finished fifth from twelve finishers, ahead of drivers including Luigi Fagioli, René Dreyfus, Ernesto Maserati, and Tazio Nuvolari. Race organiser Vincenzo Florio, begging winner Divo's pardon, called Junková the moral victor of the day. René Dreyfus later wrote: "What she hadn't considered, was that mechanical problems might interfere with her plan. Still, she finished fifth — a terrific showing. Nobody who was in that Targa ever forgot that formidable Lady."
Two months after the Targa, at the Nürburgring for the German Grand Prix, Čeněk was driving when he went wide at Breitscheid corner, struck a rock, and rolled the car. He was thrown clear but died shortly after from a head injury. Devastated, Eliška gave up racing entirely and sold her vehicles.
She channelled her love of travel into a three-month, 6,000-kilometre drive across difficult terrain to India in 1929, delivering two Bugatti Type 44 cars for Ettore Bugatti as part of a promotional campaign. In the years after World War II she married Czech writer Ladislav Khás, though Communist authorities barred her from travelling abroad from 1948 to 1964. She published her autobiography, Má vzpomínka je Bugatti ("My memory is Bugatti"), in 1972.
In her final years, following the fall of the Iron Curtain, her place in racing history was properly recognised. She attended the Bugatti Owners' Club 40th anniversary in Great Britain in 1969 and was a guest of honour at a Bugatti reunion in the United States in 1989 at the age of 89. She died in Prague on 5 January 1994, aged 93, and is buried at the Vinohrady Cemetery. On 16 November 2020, Google honoured her 120th birthday with a commemorative doodle. Prague 1 conveyed honorary citizenship upon her posthumously in 2016.