It isn't yet dusk, but the sky to the south is darkening with an approaching storm as Denis Dragusevich and Zhenya Kucheryavaya play with their three small boys in a dusty apartment-building playground in Slavyansk. When a loud rumble breaks through the air, their neighbours jokingly wonder if it is "thunder or hail" – a reference to the BM-21 "Hail" rocket launcher that government forces have fired into the city, along with howitzers and mortars.

"The three-year-olds, when the bombing starts, they point to the bathroom. They already know," Zhenya says, explaining that the family hides there during the frequent shelling. "In the future, these will be the children of war."

After pro-Russian rebels declared their own republics in eastern Ukraine in April, Slavyansk quickly became the focal point of the fighting between government forces and the steadily growing militias. Since the end of May, Kiev's "anti-terrorist operation" – even the name is repulsive to the people of the east – has shelled the city on a near-daily basis, hitting dozens of residential buildings and reducing most of the suburb of Semyonovka to rubble. One local firefighter told me his brigade works five times as much as before because of the shelling. The conflict has now killed at least 423 people, both fighters and civilians, and displaced 46,100, according to recent United Nations figures.

"Everyone talks about the same thing, about when the war will end," says Katya, who lives in a two-room apartment with her mother and her eight-year-old son, Gleb. The distribution centre where she worked has closed down, and instead she spends half the day hauling water to the apartment. In the afternoons and evenings, she sits and talks with neighbours.

"Kids draw tanks and planes when earlier they drew flowers and trees… My kid wakes up every day and asks, 'Did they bomb us?' He's scared," she says, telling me how last month they came under heavy shelling at the children's hospital where they were getting a doctor's note for Gleb's summer camp to say he was in good health.

The people I speak to tell me that life here has changed drastically: because of damage to infrastructure, water has been cut off for many since early June, 90% of the city lacks electricity, mobile reception is elusive and some neighbourhoods no longer have gas. A restaurateur named Ibragim, who was cooking pilaf over a barrel fire in a supermarket car park on a recent afternoon, said he started cooking outside after his cafe was destroyed by a shell. A general distrust of the Kiev government has become bitter hatred, and the growing cadre of rebels are widely seen as the city's saviours. Yet rather than pick up a Kalashnikov, most people still struggle to live normally in increasingly abnormal conditions. In the words of one resident: "If you get hung up thinking about how a war is going on, then it's not worth living."

Denis Dragusevich and Zhenya Kucheryavaya play with their children in the playground in front of their apartment building. Photograph: Yusuf Sayman for the Guardian

Denis and Zhenya live in a two-room apartment with her father, disabled mother, sister and their three boys, a one-year-old and three-year-old twins. The family wakes up at first light; now that there is no electricity, the day ends much earlier. After a quick breakfast, Denis and Zhenya go to haul water in plastic bottles from a nearby well. Elsewhere in the city, water towers have run hoses for lines of thirsty people to fill up.

Several times a week, they buy milk at a small cottage in the city where a man keeps a cow, pig and chickens that have been abandoned by their owners. It costs 20 hryvnia (£1) for three litres – not bad, Denis says, but more expensive than it used to be. He also sells them eggs at a discounted price: one hryvnia each.

Many products have grown pricier as food supplies have been stretched and stores have closed, with the UN estimating that seasonal vegetables are four to five times more expensive than before. The few supermarkets that have electricity are still open, but their many bare shelves betray the difficulty of delivering goods to a city under siege.

Zhenya says her biggest worry is finding food for her mother and children now that the family's income has dropped off. She previously received 3,000 hryvnia (£150) a month in child benefits, her parents received 2,600 hryvnia in pension payments, and Denis earned 50 to 100 hryvnia a day working in a tableware factory during winter – Slavyansk is known for its ceramics – and on building sites in summer. But social payments have been cut off, the factories have shut and no one is building anything during the bombardment.

"Earlier, we didn't have to scrimp on the kids' food; we bought veal, chicken, milk and sweets," Zhenya says. "Now we go home and they say, 'Momma, give me sweets', and you can't explain to a child that there's no money."

This week, the family had a rare treat when a rebel soldier gave them four kilograms of beef. She put the bones in a pot of borscht for the adults and boiled the meat for the kids. Zhenya spends the morning waiting in queues for humanitarian aid whenever deliveries get through the tight encirclement of government forces. Recently, she waited nearly five hours to receive about a kilogram of pasta, canned goods and sausage, as well as pear-flavoured soda. A tattooed biker who goes by the nickname Skull tells me he has drawn on motorcycle club contacts, including Russia's famous Night Wolves, to organise deliveries of aid, most of which comes from Russia.

Zhenya says the constant fear has changed the way she looks. "If you look at my old photos, my face has completely changed," she says.

The local Communist party has been organising assistance, from bread lines to medicine to buses to take women and children to other cities in Ukraine and Russia. Only one pharmacy is open, and prices there have risen, so finding medicine for her diabetic mother is another constant chore.

After cooking lunch, the family typically sleeps through the afternoon, while the grandfather often goes fishing. After they get up, the parents take the boys out to play – no farther than the playground in front of their apartment building, though, in case the shelling starts. In the evenings, the neighbours sit outside the stairwell entrance and listen to the news on a boombox tuned to a rebel-friendly radio station. When it starts getting dark, it's time to go in, eat a snack and go to bed, Denis says.

An apartment block in Slavyansk damaged by shelling. Photograph: Yusuf Sayman for the Guardian

The yard smells of waste because of the nearby dumpsters, which residents leave open for the many cats and dogs that now roam the city, having been abandoned by their owners. Tens of thousands have left Slavyansk, and the fruit from apricot trees in the yard is rotting because not enough residents are left to pick it. Life in a war zone is a fetid affair, first and foremost because it's hard to flush a toilet with the precious little water you've managed to haul in that day. They clean the toilet with chlorine every day, but all the same, the apartment, with its piles of dirty clothes and a bedridden grandmother, "doesn't smell like camomile", Zhenya admits.

Like many families, Denis and Zhenya have harrowing stories of the attacks they have witnessed. The most infamous of them occurred on 8 June, the Russian Orthodox holiday of the Trinity, when the government forces organised a "bloody Sunday" for them, residents tell me. One shell landed in the belltower of a church. Casualty counts varied, but both sides admitted civilians had been killed, including a six-year-old girl.

When the shelling started in the distance, Denis went to look for Zhenya, who wasn't home from her daily errands. As he was walking down Svoboda Street, he heard the whoosh of an incoming shell and dropped to the ground. A few dozen yards away, a shell hit the corner of a high-rise and two more hit the apartment building next door, sending bricks flying.

"There was no fear or surprise, just emotion, 'What are you doing, you dogs? When will it end? Bastards!'" Denis recounts.

The shelling is most often from mortars, with occasional rocket or artillery strikes, but it is the rare airstrike that everyone fears most. Zhenya remembers the day rebels shot down an An-30 reconnaissance plane: from her apartment, she could hear residents "yelling hurray and celebrating" as the news spread. Some of the few dozen residents left in Semyonovka told the Guardian the government had deployed incendiary bombs in the village this month, "lighting it up like day", although Ukraine's interior minister denied the charges.

When the shelling starts, Denis and Zhenya take the kids into the entryway of their apartment or the bathroom. The others stay with Babushka in one of the bedrooms where Denis has piled four mattresses against the windows. When shelling started around 8pm on a recent evening, he took the cushions off the couch and they slept in the entryway. Otherwise, the family sleeps all five together in one bed. One of the twins, Yaroslav, cries often and is almost always in one of the adults' arms ever since a shell fell nearby.

"When you have to give a three-year-old valerian to sleep, it's awful," Zhenya says.

Apartment building basements around the city have been repurposed into bomb shelters, including the one beneath the apartment where railroad worker Sveta Serdyukova lives with her 18-year-old and eight-year-old sons Igor and Nikita. When the bombardment is particularly strong, they sit for hours in the windowless room lit by candles and strewn with mattresses. "Life has taught us to be firm and withstand all difficulties… I have kids, I have to be an optimist," Sveta says, explaining why her family doesn't leave the city.

Katya and her son, Gleb, eight, fetch water as water has been cut off for many since early June. Photograph: Yusuf Sayman for the Guardian

Katya and Gleb try not to leave their neighbourhood in case shelling starts. They fetch water in the mornings; in the afternoons, Gleb draws pictures or plays with his action figures or his favourite Porsche Cayenne toy car. The fighting has been hard on their pet turtle and fish. Only four fish out of 12 are still alive. Katya worries about where Gleb will study in the autumn as his school was recently hit by a shell.

On a recent evening, Katya talked with her mother and several building residents about the bombings while Gleb chased a stray dog nearby. "In Afghanistan, we had an open war. This is a bombardment of peaceful people," says one neighbour. "If they want a punitive operation, they should come and fight one-on-one with the rebels."

Asked why they don't leave the city, Denis and Zhenya explain it's because of their precarious financial situation. Because of Babushka's condition, they would have to hire a taxi, and that costs 500 hryvnia (£25) for a trip to the nearby town of Svyatogorsk where thousands have taken refuge, an "unreal sum", Zhenya says.

The story is similar for other families. Gleb spent much of May at summer camp, but Katya says she doesn't have the money to send him back. "We all want to run, but where to? If I go away for a month, I'll come back to an apartment with no door and no windows," she says. She worries that her home could be looted.

Katya and Gleb left to stay with her grandparents in nearby Kramatorsk this week after their neighbourhood came under intensive shelling that hit nearby apartment buildings, killing residents. Shrapnel was sprayed across the yard where Gleb played.

"I had such a home, such a family, I was working, but the Slavyansk we knew is gone," a crying Katya said by phone from Kramatorsk. "What will we do now? Where will we go? We left with only underwear practically. They keep killing us and killing us."