Seventeen-year-old Seryozha squeezes himself through a pair of absurdly narrow bars and beckons me to follow. It's -8C, but I have to take off my coat or I'll never make it through the gap. I lie on the ground, hold my breath, and wriggle slowly under the bars. It's undignified and awkward, but there's no other way in.

The bars fence in an underground utility room that holds the heating and sewage system for a Soviet-era apartment complex. I follow Seryozha warily through a series of pitch-dark chambers, using my head torch to avoid low-slung pipes and piles of human shit. Above us, Kiev is enjoying a day of crisp winter sunshine. At the far end of the chambers, a jury-rigged lightbulb casts a pale-yellow glow on a row of filthy mattresses.

Seryozha shows me a bag of food-scraps he has scavenged from bins. The smell makes me gag. For Seryozha and at least two others, this place is home. They seem indifferent to its shortcomings; probably because they are all high. The dirt floor underfoot is scattered with empty yellow tubes. They're from a brand of Ukrainian glue that's used for resoling shoes, but when inhaled from a plastic bag, the fumes suppress feelings of cold and hunger and produce auditory and visual hallucinations. They also cause brain damage.

On one of the mattresses, a woman in her 20s is inhaling eagerly. The bag she holds clamped to her mouth inflates and deflates in time with her breath. "Look, she's sniffing!" Seryozha giggles. He has a heavily scarred face and wary eyes, but when he smiles he looks young and suddenly lovable. The woman's eyes roll back in her head and she's briefly transported to another realm. "Glukhi, glukhi!" says Seryozha. Hallucinations.

At 17, Seryozha's young enough to get a place at one of Kiev's shelters for homeless children, but he's not interested. "There are bars on the windows," he says. I point out that there are also bars on this cellar. He gives one of his cherubic grins. "But you can get out, you can go for a walk." The word he uses for going for a walk, "gulyat", means to wander around, but when street children say it, it can mean variously "wander around", "hang out", and even "beg".

Seryozha's what Ukrainians call "a social orphan": a child with one or more living parents who are unable to care for him. His is a story of alcohol abuse, beatings, spells in orphanages, and parental failure. By now, the story is familiar: I've heard a version of it from every street child I have spoken to.

He can't remember when he started using glue, and when he speaks he mumbles in a way that reminds me of a punch-drunk boxer. He has a vague plan to get into a rehabilitation centre and kick his habit, and then in the next breath he's talking about his plans for summer, and how he's looking forward to living outdoors.

No one knows how many Seryozhas are sleeping in manholes, or basements, under bridges, or on top of hot water pipes around Ukraine, but the figure is likely to be in the tens of thousands. Unicef puts the number at around 100,000, but it's a contentious estimate: it includes children who have a home, but spend a significant portion of each day on the street.

The head of Kiev's Centre for Children's Services is a charming former social worker called Nikolai Kulyeba. He says there are no more than 10 or 15 children under 18 living on the streets in Kiev. "Otherwise," he says, with a certain circularity of thinking, "I would know about it."

I list the children I have met so far: Dima, a 14-year-old from western Ukraine who sleeps on heating pipes by the railway station; Slavik, 16, who has fled from his drunken mother and lives in a basement with 12 others; Sergei, a 17-year-old with a black eye; 15-year-old Olya who is illiterate and lives in a basement full of drug addicts. I quickly get up to 11 – not including the borderline cases, such as Sabina, who says she's 18 but looks 15 and lives in a manhole. I ask Kulyeba if it's possible that in two weeks in Kiev, during a record freeze when many people didn't venture out of their houses, I might have met the city's entire population of street children. "Maybe they didn't tell you their real age. Maybe they said they were under 18 so you would feel sorry for them."

Outside Kiev's main railway station, outreach teams are offering free health checks to children aged between 14 and 17. A girl with dark hair who gives her age as 18 tentatively offers her finger to be pricked. It takes the nurse an age to coax it into surrendering a few drops of blood. Within minutes, the single lines on the test kits confirm that she is free of hepatitis B and C, and HIV, but positive for syphilis. "It's terribly bad for the baby she's carrying," says the nurse, "but, sadly, we see a lot of cases like this."

Street children are also disproportionately affected by Ukraine's HIV epidemic. The base rate of infection here is already higher than the 1% of the population that epidemiologists say is a critical threshold. The outreach teams tell me that close to 1 in 10 of all children they tested turned out to be HIV positive. Oksana, 17, says she contracted HIV when she was raped while living on the street.

She is in the early stages of infection, when no medication is necessary. But will she have access to anti-retroviral therapy when the time comes? In principle, Ukraine's constitution guarantees free health care to every citizen, but when you mention this to a Ukrainian, the reaction is a roll of the eyes, or a derisive laugh. In practice, you have to give bribes or "donations" when you visit the hospital, as well as paying for your own medicine.

Kiev's main thoroughfare, Khreshchatyk, boasts high-end stores and coffee shops, as well as a Marks & Spencer. It seems very much a part of western Europe. And yet, in some ways, Ukraine is still encumbered with its Soviet inheritance. The country has not scrapped the system of internal passports which the Soviet Union used to control the movement of its citizens. Ukrainians still have a registration stamp in their passport showing their place of residence, and while it's supposed to be irrelevant to job opportunities and healthcare, in practice, you're severely compromised without it. "Bez bumazhki, ty bukashka," goes the old Soviet saying: without the right paperwork, you're an insect.

Ukraine is spending a fortune on the Euro 2012 next month. The final will be played in Olympiskiy Stadium, high above Kiev, a gleaming piece of architecture that was rebuilt from scratch for the occasion at a cost of $700m (£440m).

"Of course, I would love to have that money in my budget," says Kulyeba with a rueful laugh. "But I'm sure it's been calculated so it will yield a profit." Across town, at one of the city's drop-in centres for street youth, the common room is full of the smell of unwashed bodies. Someone tells me that Seryozha was here earlier but was asked to leave because he broke one of the centre's rules by being high on glue. It is typically self-sabotaging behaviour: seeking help, and then disqualifying yourself from it. Three months on, the last I have heard of Seryozha is that he has moved out of his basement, but his life is unchanged in other ways. He's still begging and still using glue. He's clearly not ready to leave street life, and it's not clear he ever will be.

The director of the centre, Leonid Krysov, is bustling around trying to reunite the children with vital paperwork: the birth certificates, passports and registration papers that are a prerequisite for escaping street life.

"Usually we get an allocation, but this time I can't get any tickets to the football," he says with a sigh. "But the kids have told me not to worry. Apparently they know some heating pipes that come up right inside the stadium."

• Unreported World: The Teenagers Who Live Underground is on Channel 4 on Friday 18 May at 7.30pm.