Her name is Petra, and she is a 64-year-old live-in housekeeper. She talks of how she moved to Los Angeles from Peru over two decades ago, and of the longing she still feels for home. Today is Sunday, her day off, so she is going to the Catholic church, two bus rides away in Culver City. The Number 2 bus draws up, and she is swallowed by the soft hiss of the doors. As the bus slides by, the faces in the windows are all Hispanic or black, all weary.

The street resumes its steady composure. A red sports car hums towards the coast, and a woman in white walks in circles in the middle of Arden Drive.

This is a story of belonging and not belonging, of preposterous wealth and immense poverty; of how, in a city where people love to be seen, so many can slip through the cracks unnoticed.

It is also the story of a single street, Sunset Boulevard, a 22-mile vein that goes from the coast to the clutter of downtown, past Sunset Strip, the Church of Scientology and on through Silver Lake. And of how, if you should choose to walk that street, from sunrise to sunset, you will come to see a city unadorned and unmade, a city at odds with itself.

LA-LA LAND is a nickname that seems to have spread from the 1970s onwards, a way to capture the strange and dreamy affectations of this city. To walk Sunset is to be struck not only by the deliberately outlandish characters but by the many mentally disturbed people on its sidewalks: the woman rooting through bins who growled on approach, the man masturbating in a car park, the slink-eyed souls muttering darkly to themselves on street corners.

Then there was the peculiar encounter not far from the intersection with La Brea Avenue, as a normal-looking young man hurtled towards us on a skateboard. He was bare-chested, carrying a guitar and eating an ice cream, and it was only as he drew close that we saw something fractured in his eyes. “Save us!” he barked as he skated by. “Before they all kill us!”

And if the air soured then, it was just as suddenly sweetened by the chirruping of a man sitting among the plants on the verge, his hair a tangle of ribbons and purple plastic, swigging Bud Light from a large water bottle. “I’m in the penthouse!” he called brightly. It would be wrong to say we had a conversation. He spoke as if a string had been pulled to make him talk. Why had he come to Los Angeles, I asked, and he gave a disconcerting grin. “I’m tropical, like a dolphin!” he hollered. “You don’t put it in the snow!”

He propositioned us, and upon our polite refusal he launched into Carly Simon’s “You’re So Vain”. We all sang it, from start to finish, there on the sidewalk.

Back on the Strip, when the day was still young, we stopped by Mel’s Drive-in, a 24-hour, 1950s-themed diner offering burgers and pancake stacks and Coke floats. They were playing “Dancing in the Street” and “Beauty School Drop-Out” on the jukebox, as a waitress with bouffant hair and bright pink lipstick delivered a plate of waffles to a plump woman in a leatherette booth.

At the back, a young woman sat sketching on sheets of hotel notepaper. She was dressed in high-stacked shoes and elaborate eyeliner, and at her elbow sat a half-eaten bowl of apple pie. She frowned when I asked her name and stared for a while. The light through the window made her skin look ashen and her eyes hard. “Sun,” she replied eventually.

Sun was born in Belarus and moved to Israel as a child before heading to New York eight years ago to work in a clothing store. Somehow she ended up in Los Angeles, running away from a bad relationship and arriving with a dream of recording with Marilyn Manson. Her voice is heavily accented, her glower at odds with the California day. “I’m super-tired,” she says suddenly. “I didn’t sleep for like a week.”

She tells us she has just been released from jail, where she had been placed for trespassing at the Four Seasons Hotel. “I was just there writing lyrics,” she insists, “I was really inspired. And a gentleman…” her sentence trails off then revs up again: “I’ve been arrested twice in the past week,” she says. “The first time was because I started throwing tampons into people’s cars. I shouted ‘Free tampons everyone! Free tampons!’ I was so bored, I needed company, and some guy was walking past and I took the headphones from his ears and I told him ‘Whoo! Let’s go party!’ But he was scared and he ran away. And then the police stopped me and said ‘Are you OK miss?’ and I said to them ‘I want to drive your car!'” And then she was arrested. Now, she says, she has all these papers—she waves the court documents, squints at the small, dark print. “I wish I could throw them away. I wish I could make toilet paper out of it.”

Now she is unsure what to do. The bad relationship was psychologically harmful, she says. “It was hard for me to recover. I thought OK, my goals are acting, art, writing. But they won’t give you the papers for work. And it creates legal problems.”

She is wondering if she can stay with friends, or squat in an empty office block she has seen on Rodeo Drive, but she is more pressingly concerned by a romantic entanglement with a guy named Alex. “We met on Valentine’s Day last year,” she says. “We met at noon-time in Hollywood. I said to him ‘I want to see your eyes.’ And I took his sunglasses off and I thought ‘Oh! I love him!’ I was hypnotised. I said ‘I love your brown eyes.’ And then we went and bought contact lenses.” She sighs. “I don’t understand the differences between hanging out, dating or a relationship here in America. He started telling me he was sleeping with other girls. I wonder what the reality is? Is it a test?”

She shows us her sketches, and her notebook, full of lyrics and half-ideas, doodles and elaborate plans for an ecosystem that will cool the Earth. “I’m writing things to fix the world,” she explains. “It sounds stupid, because I don’t even have a place to be right now. I have a couple of cents left. But I feel like I have nothing to lose. When you have complete freedom, you realise you can actually survive without money and without sleep.” Sometimes, she adds, “you see the moon in the middle of the day. And so sometimes I wonder if this is for real or if I’ve been punked.”

Somewhere between Sun in the diner and the tropical dolphin of a man near La Brea lies a story of this city’s lost and lonely and weak, of how easy it is to lose your footing here, to lose your self and your sense of purpose, your job, your home, your friends, your mind. Perhaps the most la-la thing about Los Angeles is the apparent absence of a sufficient safety-net to catch the vulnerable.

THE LOS ANGELES region has one of the highest concentrations of homelessness in America. In the two years from January 2011 to 2013 the number of people living on the streets in the LA city area has increased from 25,539 to 29,682. And it’s estimated that a quarter of these people suffer from a severe form of mental illness.

Many congregate downtown, in the area known as Skid Row, with its cardboard-box shelters and shopping carts. “The poor man’s underworld,” the Pulitzer-prizewinning journalist Hal Boyle called it in 1947. “A cross-section of American futility, the place where men who have lost hope go after they have jettisoned their dreams.” But you find this same sense of futility on Sunset too: dirty bodies curled in doorways, a man asleep on a bench, his feet turtled and grey.

The climate here makes it somewhat easier to survive on the streets. It was part of the appeal for Debbie, now 58, who came here a year and a half ago from Michigan. Debbie’s story is not unusual: when the bank foreclosed on her house she decided to head west to California, “because I thought it was nice and warm.” She had fond memories, too, of a trip she made here when she was a 17-year-old hippie. She stayed in a hotel at first, but nine months ago, when the money ran out, she began sleeping on the street. For a while she moved about from day to day, but now a leg injury means that she can mostly be found tucked under a blanket outside Starbucks. “I watch people go by, I watch the traffic, I think,” she says. “Sometimes I get bored, so I take a nap.”

Some of the homeless came here with dreams of more than warm weather. A man sits smoking in a bus shelter. He is wearing blue surgical scrubs and listening to “The Essential Michael Jackson”. Calvin says he moved to LA from Houston, leaving behind a well-paid job as a surgical technician for a dream of becoming a stand-up comedian. He has a new job lined up, at a surgery centre in Newport Beach, but until then he is surviving from day to day. “I’ve been on this bench three months,” he says, jovially, “and I’ve got another month till I start work.”

The experience has been “pretty interesting,” he says. “Some motherfucker stole my food yesterday. And some homeless guys stole my money and my clothes. So this is my life.” He gestures towards his rucksack. “Two scrubs, two pairs of underwear. But the police don’t mess with me too much.”

He put off coming here for years, he says. “It’s hard to leave a good job, good money…but my heart kept calling me to LA.” Now 40, he has wanted to be a stand-up since he was 15 and watching “Showtime at the Apollo” on TV. This year, he even performed at the Oscars after-party at the Comedy Store. “You hold on to your dreams,” he says, and it is only later that I begin to wonder if any of it was true.

But still they come. On our walk we meet aspiring directors, actors, cinematography students, trainee music producers who dream of writing soundtracks, wide-eyed young men from Sacramento queuing for brunch outside the Griddle. We meet Matt and Bill, smoking cigarettes outside the Guitar Center. They live in the Valley but came here for the weekend to record with their “sorta nu-metal, thrash-metal” band. “Friday was a good night,” says Bill. “We got wasted. We had the cops called on us 12 times. And we had to shower our guitarist because he threw up in his hair.”

In the summer of 1950, Paramount Pictures released “Sunset Boulevard”, the story of an aspiring screenwriter who heads to Los Angeles from Ohio and ends up dead in a swimming pool. It is a cautionary tale, one that encompasses a pet chimpanzee, a faded movie star, and the dangers of both delusion and aspiration. It culminates, famously, with the washed-up actress addressing a great director: “Alright, Mr DeMille, I’m ready for my close-up.”

MOST PEOPLE DO not see Hollywood close-up. They see it on their TV screen or in gossip magazines. Even if they come here, they see it through the car window. They see it only to the soundtrack of their car stereo, feel it only in the thud of warm air between air-conditioned vehicle and air-conditioned coffee shop.

When we mention that we are walking the length of Sunset, people look at us in disbelief, assuring us that it was not only dangerous but most definitely weird. At street level, though, you see more: an IBM laptop in a discarded takeaway box holding seven prawns; two men dancing in the back room of a salsa club; the words “Love Is What You Make It” scrawled across a wall. You catch the faded incense as you walk past the Church of the Blessed Sacrament, see Jayne Mansfield’s pink suitcase displayed in the window of the Dearly Departed Tours Office and Curiosity Shop, with a sign beside it instructing you to “note the damage”.

You meet people like José, a former labour-union worker who, six months ago, opened a taco van here on Sunset. “I’m 52 years old,” he smiles, “but this is not a midlife crisis. I am halfway through my life and it is a checkpoint: where are you in life? Where you want to be? I consider this a journey. There’s a very bright light at the end of the tunnel. I can see it.”