"They call me Fearless Felix," says the man who, with nonchalant courage, fell to earth faster than the speed of sound. Less than three weeks ago, Felix Baumgartner reached an altitude of 128,100 feet in a small capsule attached to a helium balloon before he plummeted back down again through 24 miles of cold blackness at a top speed of 833.9 miles per hour.

His space jump was watched live on YouTube by more than eight million people, and the fevered reaction online was matched by saturation coverage in the traditional media. As a curiously driven man, who had dreamed of flying ever since he was a five-year-old boy in Austria, drawing detailed pictures of himself soaring through the sky, Baumgartner had achieved his greatest ambition. He had moved from the often-illegal activity of base-jumping – having blagged his way past lax security at some of the world tallest buildings so his daredevil talents could be noticed – and become one of the world's most celebrated men.

Baumgartner sits in a plush chair in a London hotel and arches a wry eyebrow at his cartoonish nickname. "You and I know Fearless Felix doesn't really exist," he says, quietly, and more thoughtfully than might be expected. "He might seem like a cool guy, but I've had to address a real psychological battle. It's been way harder than stepping out into space."

He may be a certified celebrity, with an American twang to his Austrian accent, but he talks with the zeal of an ordinary man who has just survived an extraordinary experience. Baumgartner also uses the very human confines of psychological frailty, rather than the vast expanse of space, to frame his achievement. A canny publicist, the 43-year-old is smart enough to recognise that there is real strength in admitting moments of weakness. But there is also something surprisingly moving in his revelations that the source of his suffocating fear was an old-fashioned spacesuit.

"I feared and hated the suit because of my desire for freedom. I started skydiving because I loved the idea of freedom. But you get trapped in a spacesuit, and people are adding weights to it every day.

"They'd say, 'Right, we need oxygen bottles,' and then a couple of weeks later it would be: 'You need a chest bag.' That chest bag became bigger and bigger and the suit is twice my normal weight. Skydiving is now no fun at all. It's scary. I remember my first dive with this suit. I was standing at the exit at 30,000 feet, and it felt like my very first skydive. The same fear from 25 years ago is back. It never felt good in that suit because it never became a second skin.

"Normally, when I skydive, even in winter, I wear very thin gloves. I want to be flexible, with fast reactions. But a spacesuit slows you down. You have big gloves and you cannot move your head very well. A natural movement, when you pull your chute, is to look up. But with the suit you cannot do this. So now I have two mirrors on my gloves.

"You open the chute and you look down at the mirror to see if it's fully inflated. Every skill I had developed over the years became pretty useless as soon as I stepped into the space suit. And after 25 years as a professional, it makes you feel weak and exposed."

Baumgartner cut a lonely figure as he prepared to leap into exultation or oblivion. But his vulnerability was bound up in claustrophobia. "I only started getting anxious if I was in the suit more than an hour. You can fight your way through an hour. But if it takes five hours you're never going to win that battle. So that's why I had to address it."

The problem became so distressing that Baumgartner required psychiatric help. "This is the first time I needed [psychological] help," he winces. "It was so embarrassing in the beginning. They'd say things like, 'How would you describe what happened, to your son?'"

He scrunches up his face. "I don't have a son. So I didn't feel like talking to my imaginary son. But at the same time, I thought: 'If it gets rid of my anxiety, I'll talk to my invisible son.'"

Suddenly a sombre voice booms out: "Attention please, this is an emergency."

Baumgartner's eyes widen. "Is this real?" he asks.

"Please leave the building now by the nearest available exit … do not use the lifts or escalators."

Fearless Felix and I sit tight and keep talking. Joe Kittinger, the 84-year-old who set the previous record for the highest space jump of 19 miles in 1960, and became Baumgartner's chief adviser, pops his white-haired head around the corner. "It's a real emergency," he says.

"We're relaxed," Baumgartner quips.

"Good luck," Kittinger says drily as he moves towards the exit.

"Hey Joe," Baumgartner shouts, "see you in heaven … "

He returns to his story. "Can you imagine how embarrassing it is to have two strangers listening to you talk to your invisible son about your deepest fears?"

Even the smell of the suit unsettled him. "It was the smell of rubber. That was always the key moment. But my anxiety started the day before. I would not sleep well, and then you have to drive to Lancaster [in California, where his test capsule was based]. When you get over that last hill you can see Lancaster down there … and you know the suit is waiting. The psychiatrist called it the 'train of negative thoughts'. I was always riding that dark train. He said you have to get off it with positive thinking. It's easy to say, and hard to accomplish. Still, we did it. I started to feel strong again."

Yet Baumgartner had walked away from the project for six months. It was only when he saw footage of a replacement doing his job in testing that he was shocked into returning. "I felt jealous," he says, "and I thought: 'You're not supposed to be in my suit.' I saw the BBC film yesterday [Space Dive, which airs on BBC2 on Sunday] and it's disturbing because you see my name on the helmet, then you realise: 'Hey, that's not Felix. Some other guy is in there.' No offence to Rob [the test pilot]. We had this big test and they couldn't say: 'Hey, we'll skip it because Felix is too weak.' But it really hurt my feelings when I saw Rob in my suit. It felt like I'd been replaced. Of course, it was part of the journey, but when you're inside that situation you never like drama."

As if to remind us that drama can happen anywhere, a representative of Red Bull, the Austrian company that spent over $18m to help Baumgartner fulfil his dream, urges us to follow her.

"If you see fire, tell us," Baumgartner says laconically.

"This is real," she says, calmly.

"OK," Baumgartner laughs. We leave the hotel and Baumgartner keeps talking, intently. "The toughest moment was when I lost my team after I came back from Austria," he says. "My psychiatrist told me: 'Nobody thinks you can do it anymore. You have to get your leadership back.' I went into this room and I could see everybody sitting on the other side of this table. All my friends. And just by the body language I could tell: 'Nobody thinks I can do it anymore.'"

Did Joe doubt him? "Everybody," he says sadly. As a former soldier and a self-proclaimed team-leader and man of action, Baumgartner was shaken by the loss of faith in him. "Art Thompson [the project director]. Mike Todd [his life support engineer]. I never thought Mike would doubt me because he was like my father. He was the key guy in those quiet moments when he was dressing me in the locker room – like a boxer with his coach before he goes to fight. But he was sitting on the other side now. Nobody had faith in me anymore. That was a really bad moment. This claustrophobia was the only weakness I had. It's not my fault. It's just in my mind."

Felix Baumgartner sits in his capsule during preparations for the final manned flight of the Red Bull Stratos mission in Roswell, New Mexico Photograph: Joerg Mitter/AP

Baumgartner's words are poignant rather than plaintive – but he sounds like a sports jock when describing the "game plan" and "strategy" he developed to regain control. "I thought, whatever it takes to get my leadership back, I'm willing to do it. After five days it was working. Two weeks later, everyone was positive and we knew I was ready."

Doubt, however, still plagued him. "The worry is I won't fly supersonic or, in the worst-case scenario, I'm not as fast as Joe Kittinger was in 1960. You have to explain to the world that, 52 years later, you're slower than Joe? It's another pressure. I don't think people get what it means to do something when the whole world – from the pope to the president of the US – is watching you."

His problems continued. As he floated towards the 24 mile-high mark – it took almost three hours for the balloon to lift his Red Bull Stratos capsule into space – Baumgartner's visor began to cloud as he exhaled. The prospect of doing the jump "half-blind" threatened the mission, and he had to endure various tests before it was established that his equipment was working.

He had less time once he had left his capsule, and begun to whirl through space at a speed which sent him into an inevitable spin. "I had one minute to find a solution. While spinning, I'm thinking: 'Should I push the button to release my drogue chute, to stop that spin? But that would mean it's over and I'm not going to fly supersonic – so should I tough it out and find a solution?

"I had to maintain my cool and this is what I've been doing the last 25 years – being focused and not freaking out. In my head I was cool-minded. My worst fear was not dying, but failing to fly supersonic. If you're at 3.6 Gs for six seconds, it fires the drogue chute. I was rotating, but it was hard to tell how many Gs I was at. I felt I had it under control and, hey, I'm not dying. But I couldn't know how close the drogue chute was to firing. In the end it was OK, but it was difficult, and that's why Joe held the record for 52 years. Lots of people underestimated it."

Felix Baumgartner jumps out of the capsule on his way to breaking the world record for the highest free fall in history. Photograph: Handout/Getty Images

Baumgartner thinks hard when asked about his jump's sweetest moment. "I had a couple of good moments," he eventually says. "One was standing with my feet outside the capsule just before I stepped off. We'd been working towards that for five years. As soon as I was standing there – completely released from all the cables – I knew it was going to happen. That was a big relief and a really unique, outstanding moment.

"And then when you open your parachute you know it's over – I'm still alive! Mike Todd was the last person I saw before going up. He'd said, 'OK, see you on the ground, buddy.' But you could tell he wasn't 100% sure. I wasn't either. We prepare for the worst but hope for the best. And then, three hours later, Mike is the guy I see first after it's all over. Mike worried about me like I'm his son. But when he's happy he looks 16 again. I was looking forward to seeing that smile."

This is Baumgartner at his most likable; he is also touching when describing the emotional toll on his mother, Ava, whose sister was buried just a week before he made his jump. His life is not always simple and next week his lawyers will appeal his conviction for punching a lorry driver during a traffic jam in 2010. And so how will a man consumed by outrageous challenges rekindle the intensity of his space jump? "I don't have to," he replies, confirming his plan to become a rescue-helicopter pilot. "I reached a peak and I don't have to top it again. A lot of kids now think of me as Fearless Felix – but I hope I can make fear cool. All these kids can know that Felix also has fear. So they can address their own fears. I did it – at first I would consider the suit a handicap. And handicapped people have to find a way to live with their handicap. The suit was my worst enemy, but it became my friend – because the higher you go, the more you need the suit. It gives you the only way to survive. I learned to love the suit up there. That's an even bigger message than flying supersonic."

Space Dive is on BBC2 on Sunday at 8.30pm