A small, fragile-looking woman with “Super Diva” stamped on her sweatshirt is stretched out over an exercise mat in the gym, her face locked in a rictus of concentration as she does 20 push-ups. “She is like a cyborg,” says her personal trainer, keeping watch over his client as she goes on to hold a steely plank then perform a series of bone-rattling squats. “And when I say cyborg, she is like a machine.”

At 85, having already survived bouts of colorectal and pancreatic cancer, Ruth Bader Ginsburg is the proverbial survivor. And she needs to be. There is a very great deal hanging on her continued longevity as the fourth liberal-leaning justice among the nine justices of the US supreme court.

As such she provides the last line of defense against the forces of darkness that are never far beneath the surface in American public life. Not least the current incumbent of the White House whom she famously denounced before his election as a “faker” (she later apologized). Should Ginsburg vacate her seat while the faker is still in office, she would give Donald Trump the golden opportunity to lock a conservative majority into the nation’s top court for at least a generation.

In the new documentary RBG, directors Betsy West and Julie Cohen seek to answer the mystery of how such a quietly spoken and restrained person not exactly renowned for her effervescence – her children kept a record when they were little of the very rare occasions she was humorous called Mummy Laughs – morphed into a hipster icon likened to fellow-Brooklynite, the late rapper Biggie Smalls.

“We were taken by this juxtaposition of this very tiny, soft-spoken 85-year-old grandmother being tough, speaking truth to power and, yes, even doing planks and push-ups. The whole unusualness of that combination is what makes her a star,” Cohen told the Guardian.

The film-makers spent the best part of three years following Ginsburg around the country as she gave lectures and attended her beloved opera, as well as conducting interviews in her chambers in the supreme court. It was a daunting prospect at first.

“Despite her small stature, Justice Ginsburg is a very intimidating person. We absolutely found it intimidating initially – there isn’t a lot of small talk,” West said.

But over time the relationship loosened. At the end of an interview at the court, the directors started playing a clip of Kate McKinnon impersonating Ginsburg on Saturday Night Live. The real life Ginsburg, watching herself being parodied on the small screen, collapsed into a fit of uncontrolled giggling that is quite startling to watch in the movie.

The film marks the 25th anniversary of Ginsburg’s nomination to the supreme court on 14 June 1993 and of her confirmation two months later with the astounding backing of 96 senators to three – a display of bipartisan support that seems unthinkable today. One aspect of her career the co-directors were particularly keen to explore, though, was the lesser-known story of her seismic work on women’s rights that consumed her long before she became a member of the highest court.

“She would have earned a place in history even had she not become a supreme court justice,” said West. “She changed the law of the land for American women, making sure the US constitution applied to men and women equally – that’s a tremendous legacy.”

As the documentary relates, in the 1970s Ginsburg played a leading role as a legal warrior for women’s rights. She was to gender equality what her predecessor on the supreme court bench, Thurgood Marshall, was to race equality in the 1960s.

Ginsburg argued six gender equality cases on behalf of the ACLU in front of the supreme court justices – all nine at that time were male, and most of them oblivious to the visceral discrimination endured at that time by women in the workplace. Despite those prevailing prejudices, she turned them round and won five cases out of the six.

She did so forever with the same poise and style – politely, respectfully, but always forcefully – that has become her trademark. Ginsburg says in the film that she inherited her composure from her mother. “‘Never in anger,’ my mother told me. That would have been self-defeating. I did see myself as a kind of kindergarten teacher in those days, because the judges didn’t think sex discrimination existed.”

But there was also an unmistakable fire in her belly. You can see it most clearly at the point in the movie when the directors get her to recite the famous words she spoke in 1973 when she presented her first oral arguments to the supreme court in Frontiero v Richardson, a case opposing unequal benefits for married women in the air force.

“I ask for no favor for my sex,” she said, quoting the abolitionist Sarah Grimké from 1837. “All I ask of our brethren is that they take their feet off our necks.”

Ginsburg enunciates each word on camera with the same fearsome focus with which she does her push-ups in the gym. The impact is electric, and the message unmistakable: underestimate me, or my cause, at your peril.

For RBG, the issue of rampant sexism was not just a legal challenge, it was personal. She began to train in the law in the 1950s when the profession was considered by most (male) practitioners to be unsuitable for women.

She was one of nine women in a class of more than 500 men at Harvard law school, where she was refused entry by the Lamont Library because of her gender. The dean of the school graciously invited all nine female students to dinner, then asked each of them in turn to justify taking a seat that could have been occupied by a man.

After graduating from Columbia University in 1959, not a single New York law firm would employ her. She was turned away by partnerships that today she employs to do research for her as a US supreme court justice.

“The idea was,” she tells the film-makers, “that men were the breadwinners that counted, and women were pin-money earners. So women woke up and complained.”

“Woke up and complained” would be a very good description of Ginsburg’s approach to serving on the country’s highest court over the past quarter century. As the legal and ideological fulcrum of the court has moved to the right over those years, so Ginsburg has become more pronounced in her opposition to the majority’s rulings.

An RBG dissent has become a jurisprudential wonder to behold. In 2006 in Ledbetter v Goodyear, a case over unequal pay for women doing exactly the same work as their male colleagues, she challenged Congress to change the law and it duly did, drafting the Lilly Ledbetter Fair Pay Act that became the first bill to be signed by President Obama.

When in 2013 the conservative justices tore a bloody hole in the Voting Rights Act that for almost half a century had protected African Americans from racist disenfranchisement, she issued another withering dissent. She took the chief justice, John Roberts Jr’s, argument in Shelby v Holder that America had “changed”, no longer requiring exceptional means to combat racism, and sent the five conservative justices who had voted in the majority straight back to kindergarten.

She said: “This court’s decision is like throwing away your umbrella in a rainstorm because you are not getting wet.”

Though they began filming RBG before the explosion of the #MeToo and Time’s Up movement, the film-makers see their work as forming part of a wave of recognition of the extraordinary contributions made by exceptional women. “There’s a long history in this country of women’s stories being undertold, but now there’s understanding that we’ve only been paying serious attention to half of the population,” Cohen said.

Both directors feel privileged to have spent time with Ginsburg, especially in the gym filming her push-ups. “That was reassuring, certainly,” West said. “I was relieved to see that she has an energy level beyond anything I’ve seen in a human being – and that includes women decades younger than she is.”

• RBG is out in the US on 4 May with a UK date yet to be announced.