NASA's reply was simple and definitive: No girls.
"It was the first time I had hit an obstacle I couldn't overcome with hard work and determination, and I was outraged," she would write in her book, "Living History."
More than a half-century later, and after much hard work, much determination, and most of all, many, many obstacles - some undeniably of her own making - Clinton is no closer to actual space travel. She may have to settle for becoming the first female leader of the free world.
Her journey - more than three decades in the public eye, and counting - has been unlike any seen in American politics: a story of great promise, excruciating setbacks, bitter scandal, stunning comebacks, and especially reinvention - of her own life, and as a result, of the role of women in government.
Think about it: Is any woman more recognizable on a global scale than Hillary Clinton? If Barack Obama was the presidential candidate who seemed to come out of nowhere, Clinton's the candidate who seemed to come out of everywhere.
Americans first knew her as a governor's wife and working mother in Arkansas, then as the nation's first lady - famously claiming an office in the West Wing of the White House, not the East, as half of husband Bill Clinton's "Buy one, get one free" bargain.
Touched by scandal from Whitewater to Monica Lewinsky - but also carving out her own political identity - she emerged to become a hard-working senator, the first first lady to gain elected office. We knew her as the presidential candidate who suffered a stinging defeat to Obama in 2008, but proudly claimed "18 million cracks" in the glass ceiling.
Then she reinvented herself again, becoming Obama's secretary of state, traveling almost a million miles to 112 countries. Finally, after much speculation, she announced her second run for the presidency.
We knew her so well by then.
Or not. Who WAS Hillary Clinton, and why, if we'd been watching her for so long, did we feel like we didn't know her?
At least, that's the persistent narrative. Perhaps it's a question of layers. She's had so many different roles, of course we've seen different facets of her. But there's also a sense of impenetrability, exacerbated by her penchant for secrecy - a characteristic that has led to her greatest vulnerability in this election: the email scandal over her use of a private server.
For the last 14 years, and 20 overall, Americans polled by Gallup have named Clinton their most admired woman in the world. But consider some other titles attached to her over the years: Lady Macbeth. Washington insider. Robotic. Wildly ambitious. Congenital liar. (Or Donald Trump's current favorite, "Crooked Hillary.")
But also: Feminist heroine. Glass-ceiling breaker. The most prepared in the room. The most qualified presidential candidate ever. Loyal friend. Witty companion. Mom. Grandma.
"It's an amazing life," says biographer Carl Bernstein, who wrote a 600-page book on her and says he still struggles to define her. "You could not make any of this stuff up."
There have been polarizing figures in politics before, but it's hard to imagine any have been called as many things - wildly divergent things - as she. Does everyone simply have their own version of Hillary Clinton?
THE AMBITION THING"Saturday Night Live" has been turning out versions for a good 25 years. Each actress spoofing Clinton - there have been nine, including Miley Cyrus rapping in a bandeau - has put her spin on the part. But there's been one constant: ambition, pure and unadulterated.
"No, MINE!" blurted out Amy Poehler's Hillary, alongside Tina Fey's Sarah Palin, in agonised disbelief that John McCain's running mate was still in the race but she wasn't.
Recently, Kate McKinnon has perfected a wackier, more manic ambition. In a recent scene where Clinton herself gamely played a bartender, the fake Hillary asked Huma Abedin, her aide, "Why won't the people just let me LEAD?"
Comedy aside, the ambition tag has dogged Clinton, 68, throughout her career, as if it were a bad quality rather than a necessity in high-stakes politics. The satirical website The Onion captured the irony in a 2006 headline: "Hillary Clinton Is Too Ambitious To Be The First Female President."
That gets a knowing laugh from Melanne Verveer, Clinton's chief of staff from her first lady years.
"If a guy is described as ambitious, it's a noble attribute - he wants to put himself ahead," says Verveer. "But if a woman is ambitious, it's not an attribute, it's a negative, a pejorative. It's not proper somehow."
Former Rep. Patricia Schroeder thinks the ambition factor is - unfairly - key to Clinton's challenges connecting with the electorate.
"We still don't like a woman who is showing ambition, especially for that level of a job," says Schroeder, who famously explored her own presidential candidacy decades ago. "It's: 'I'd like her if she weren't so damned ambitious. How come she wants all that power?' "
CHAMPION FOR WOMENAt her college graduation in 1969, Hillary Rodham was already blazing a trail: The senior from Park Ridge, Illinois, was the first student chosen to address a Wellesley commencement. She delighted many classmates when she delivered an on-the-spot rebuke to the previous speaker, a U.S. senator whose comments the grads found condescending to women. At Yale Law School, where she met Bill Clinton, she developed a keen interest in children's rights, which she pursued in post-graduate work.
It's been a particular frustration to Clinton's campaign that young Democrats haven't responded more enthusiastically, with many attracted to the populist message of Bernie Sanders (six years her senior). There's a sense that millennials are too young to remember her efforts on behalf of social justice, particularly for women and girls on a global scale.
"Young people today want to be part of something bigger ... but they don't understand how much she shares those aspirations of theirs," Verveer says.
A key moment in Clinton's political journey - and a defining personal moment - came in 1995, when as first lady she spoke at a U.N. Congress on women in Beijing, declaring, "Human rights are women's rights, and women's rights are human rights."
It was a time when Clinton was searching for a new identity, having failed to reform health care back home. But even she had no idea the impact those simple words would have.
"It not only gave her an instant sense of the world looking at her differently, but she was also seeing the role she could play - in ways perhaps she had never understood before," Verveer says. "It has remained with her ever since."
Clinton's image as a champion for women has been complicated by her, well, complicated marriage - she's been an object of both sympathy and blame for staying with her husband post-Monica Lewinsky.
But memories of Beijing endure.
To this day, Verveer says, people come up to Clinton on her travels and say: "'I was there, in Beijing.' It's something that they instantly share."
ROBOTIC OR HUMAN?Part of the narrative on Clinton has been her trouble connecting to the public. "I am not a natural politician, in case you haven't noticed," she said recently, "like my husband or President Obama."
One "SNL" skit has her showing off her new kitchen in a Senate campaign ad, saying in robot-speak, "I can't wait to prepare some food dishes in this kitchen, such as salads and toast."
Those who've watched her up close say she's both natural and an excellent communicator one on one. Friends always say she's relaxed, funny, witty, a great companion.
And not just her friends. Talk to classmates from Wellesley, even those who only knew her from afar, and they say they can't understand the disconnect between public and private Hillary.
Nancy Herron, who didn't really know Clinton at school, reconnected with her decades later at a reunion, where Herron performed a standup routine on what it's like being in the shadows of such a famous classmate. She even skewered Clinton's pantsuits.
"She sat there and just laughed her head off," says Herron. "She really enjoyed being teased. Afterward, she gave me a hug, and said, 'We need to take you on the road!'"
Adds another classmate, Cheryl Lawson Walker: "She wasn't intimidating, easy to talk to, very funny, hang-loose. She had yet to be hardened." Or, Walker allows, maybe the "hardened" Hillary is simply what the public sees.
THE TRUTH ISSUEIn a February Gallup poll, the most common responses Americans gave when asked what came to mind about Clinton were "dishonest" and "dislike her." (For Sanders, they were "socialist" and "old.")
Fair or not, it's a theme woven into the Clinton story - both Clintons - from the White House scandals to the email story.
"The most difficult thing Hillary Clinton has to deal with right now is her difficult relationship with the truth," says Bernstein, author of "A Woman in Charge: The Life of Hillary Rodham Clinton."
Bernstein is quick to point out that Clinton's version of untruthfulness is in a different ballpark than that of Trump, who, he feels, "just spouts lies, and has no interest in existential truth," where Clinton "tries to establish a baseline of truth." (Politifact, the fact-checking organization, asserted at one point that 27 percent of Clinton's statements it investigated were false or mostly false, compared with 76 percent of Trump's.) Another longtime observer, writer Gail Sheehy, attributes her difficulties with the truth to a defense mechanism honed over years of fending off attacks on her and her husband.
"You could call it denial," says Sheehy, author of "Hillary's Choice." ''It's a defense mechanism she has used a great deal."
The issue has never been more important than in this campaign, when both Clinton's veracity and judgment are being called into question.
What the email mess shows, Bernstein says, is "this fierce desire for privacy and secrecy that seems to cast a larger and larger shadow over who she really is."
REALLY ... WHO IS SHE?Who she really is. There's that question again.
Is it a fair one? One we'd ask about other candidates?
Schroeder thinks not. "I say to people, 'What more do you want to know?'" the former congresswoman says. "We can see her voting record. We know what colors she likes. She speaks about her mother. She's a Methodist. How many politicians do we even know that much about? Do they want some kind of a confession?"
Others note that Clinton has naturally become very guarded, given that she's been judged, relentlessly and often unfairly, "on a huge stage, for all of her life," in Bernstein's words. Besides, "too many people are interested in looking for information that reinforces their already held prejudices and beliefs," he says.
Herron, Clinton's college classmate, feels that we don't subject male candidates to the same scrutiny, always looking for another layer. "What do we know about Mitt Romney? What do we know about ANYBODY? We expect her to let her hair down, to talk about her failures and self-doubt or something.
"You know what, she's not like that! Let her be who she is."