Laurie Penny can certainly coin a phrase:

"A woman's opinion is the mini-skirt of the internet," "Of all the female sins, hunger is the least forgivable; hunger for anything, for food, sex, power, education, even love" and "You can't save the world one man at a time." Although, as the radical left-wing journalist, polemicist, activist, prolific blogger and self-described "reprobate," points out, "That doesn't stop many of us trying."

Penny, who is 27 years old, mini-skirted, slightly-built and elfin-featured, with hair the colour of purple ink, can be trying, too, in some people's view. Mainly because she's scrappy and opinionated, particularly about politics.

At the age of 23, she was the youngest writer to be shortlisted for the Orwell Prize for political writing for her blog Penny Red. In 2012, she won the British Media Awards' "Twitter Public Personality of the Year" prize, which she accepted in absentia with a terrific speech about the ongoing harassment of women on social media, a subject on which she's something of an expert and on which she'll debate with another expert on the subject, classicist Mary Beard, in London later this month.

As Penny's 93,000 Twitter followers will testify, you either love her or loathe her. Irvine Welsh is a fan, as is Channel Four broadcaster and writer Paul Mason, one of her mentors. Still, the inevitable epithets - "Penny Dreadful" and "Bad Penny" - cling like iron filings to a magnet. Far worse insults have been hurled at her, from bomb threats to promises of appalling physical abuse: violent rape, torture, burial in a shallow grave ...

"I have learned just what a fearful thing it still is to be female in public life," she declares.

"Penny Dropped," was the gleeful 2012 headline reporting that she was leaving the national newspaper where she'd been employed as a columnist and journalist. Like the proverbial bad penny, however, she always turns up somewhere else, because she writes well, incisively and cogently with passion, even if she is given to the sweeping generalisation.

Currently, she's a columnist and contributing editor at the New Statesman and editor-at-large for cult New York literary project, the New Inquiry. From time to time, she pops up on Newsnight or Channel Four News crossing keyboards with the likes of Anne McElvoy, the Economist's public policy editor. Nonetheless, Penny insists that she doesn't earn a pretty penny for her thoughts, although she writes for Vice, The Nation, and various other "dead-tree media" here and in the States.

Next month she vacates the London house she shares with nine others to become an international fellow at Harvard University, where she'll join the class of 2015 at the prestigious Nieman Foundation for Journalism.

"I'm SO VERY excited!" she exclaims, clapping her hands when we meet over coffee in a Soho cafe.

She's also "very, very excited" about the publication of her fifth and latest book, Unspeakable Things: Sex, Lies and Revolution, in which she writes: "This is a story about how sex and money and power put fences around our fantasies. This is a story about how gender polices our dreams." It is a feminist book, a serious-minded one.

"I cannot force a smile for you," she warns. "As a handbook for happiness in a f***ed-up world, this book cannot be trusted." But then you could say that about all of Penny's previous publications, which include word-of-mouth bestseller Meat Market: Female Flesh Under Capitalism; Penny Red: Notes from a New Age of Dissent; Discordia: Six Nights in Crisis Athens (co-authored with Molly Crabapple); and the excellent Cybersexism: Sex, Gender and Power.

As in all her work, she weaves personal stories and interviews into Unspeakable Things, which has a punkish black cover with a bronze-embossed zip. She writes in graphic detail about her teenage anorexia and her treatment in a psychiatric unit -she's very good on the pressure on girls to appear perfect at all times, as well as on mental health issues, a subject she intends to investigate further.

She describes how she tossed her "virginity aside like ballast," tells how she confronted a man who raped her, and how when she fell for a guy at an Occupy protest, he announced that he "only kisses pretty girls."

On the comments section of a website following politicians and journalists, it's been proposed that she be stoned to death - "my favourite would be a Public Hanging or Decapitation ... Perhaps she should be Circumcised, only sew up her mouth." She's been trolled and stalked, sometimes afraid to leave the house, eventually calling the police and moving to a safe house. A photograph of her head has been pasted on pornographic images and posted online, alongside cartoons of her being beaten up. She's even received phone calls whispering about her sexual history.

Germaine Greer, she notes, once wrote that women had no idea how much men hate them. "Well, now we do," says Penny.

The daughter of Lib Dem-voting lawyers - her family tree has Jewish, Irish, Maltese roots - she was born in London and raised in Brighton, although she says she actually "grew up on the internet." She has a degree in English from Wadham College, Oxford, but insists that she comes out of the internet TMI tradition [Too Much Information], which is why she shares her own story, as well as those of others she interviews.

"I cut out a lot of stuff about myself that I felt wasn't necessary," she says. "Bur for me, the politics of the personal are important. Yes, the personal is political, but that doesn't mean the political always has to collapse into the personal. The stories I tell - about 'lost boys' or sex workers, say - aren't randomly chosen. These and my own story tell how I came to the political position I hold."

Despite her talent to annoy, doesn't she ever want to climb out of the bear pit of brutal online spats and cyber bullying by trolls? Isn't it terminally exhausting as well as scary?

"Well, I want to be accountable. I could just say, 'F*** it!' but that's not how I feel I can progress as a writer and a thinker. There has to be a dialogue if you campaign, as I do, against sexism and for feminism, as well as LGBT rights, civil liberties and politics. However, I don't have a thick skin. It's stressful. I find it difficult being on the receiving end of hatred. But if you ask me whether I'm an activist first and a writer second, I'm a writer. Writing is what I care most about, telling true and powerful stories, although I'm working on some fiction as well.

"My wonderful father, Ray Barnett, passed away very suddenly in September. I hadn't written fiction since I was in my teens but suddenly I've been making up all these stories. Real world politics didn't seem to matter much for a while obviously. I haven't shown my fiction to many people. But I miss my dad a lot. He was a kind, gentle man. It's been hard for all of us - I'm the eldest of three girls. We're very different. I'm so lucky to have smart, beautiful sisters," she says, showing me photographs of the trio on Father's Day last month.

"My father was very proud of me, though I'm not sure he understood the politics. I think he liked the fact that I was getting on, that Jewish thing that you make a name for yourself professionally.

''We were so close that when he died, I felt I might go mad again. I ran away to New York and Berlin. Mourning makes you crazy. But I have to stay well and healthy - I'll always be a recovering anorexic."

By far the best chapter in Unspeakable Things is "Love and Lies". When I tell Penny this, she says it's her favourite, too. She confesses in it that some days all she does is write love letters and that she began to pay attention to the art of writing such letters by sending handwritten mail to friends in prison. Now, she writes love letters in 10-word Twitter messages and 20-page emails to friends and former lovers across the world. I wish, however, that she still wrote some in that purple "ink" she poured over her urchin crop.

"Oh, but I really am committed to my online work, to my more ephemeral work, because I think that's the way that change really happens now," she responds, inhaling an e-cigarette. "With digital media you can have several conversations going at the same time. I can be fighting with someone in one window on Twitter; on another I might be writing a massive manifesto; on yet another, I could be organising a demonstration. That's how I almost always spend my days. I was built for the internet. Honestly, I believe that the geek shall inherit the earth."

With her unswerving beliefs, fierce desire to change the world and predilection for behaving badly, ukulele-playing Penny is surely the embodiment of American historian Laurel Thatcher Ulrich's provocation that "well-behaved women seldom make history."

Let's hear it for "young, lady writers" behaving badly.

Unspeakable Things: Sex, Lies and Revolution, by Laurie Penny (Bloomsbury, £12.99).