By Yaroslav Trofimov

Krystyna Krayevska came to Kyiv from Poland, where she normally lives and works, for her niece Darynka’s sixth birthday in January. A few days later, Darynka was diagnosed with a brain tumor and, after complications following surgery, now lies on life support in Ukraine’s largest children’s hospital, Okhmatdyt.

A clash just outside the medical facility between advancing Russian units and Ukrainian forces over the weekend left bullet holes in some of the hospital’s windows. Darynka, unable to breathe on her own, remains in the pediatric intensive-care unit upstairs. Ms. Krayevska spends her days on a cot in a basement that has been converted into a bomb shelter, next to other parents and children who could be moved to its relative safety.

“I live in fear, but not for myself. Every morning I wake up, cross myself and pray that nothing hits the ICU room,” she said as yet another air-raid siren rang out, warning of an incoming Russian airstrike on this ancient city. “She is fighting for her life, up there, and we down here are fighting for our own lives, thanks to the Russian soldiers.”

Kyiv hasn’t experienced the kind of indiscriminate shelling of civilian neighborhoods that devastated the country’s second-largest city, Kharkiv, on Monday and Tuesday. But as Russia pours new forces into Ukraine in its attempt to capture the capital and decapitate its democratically elected government, doctors here are bracing for an influx of casualties, children and adults alike. An entire area of the hospital complex, which usually caters to 20,000 children a year, has already been converted into an emergency trauma ward.

“It’s just so vile,” said Okhmatdyt surgeon Volodymyr Vovkun. “We hope it won’t happen, but we are getting ready for the mass bombing of civilians here, too. We are watching the news and know that the situation is getting worse.”

Disruptions caused by the war are already exacting a heavy toll, said the hospital’s director, Volodymyr Zhovnyakh. Diabetic children usually served by the hospital can no longer get insulin, and others have no access to baby formula. Some 10 children a day used to be operated on in Okhmatdyt for appendicitis every day, compared with one a day now, he added.

“These children are still here, but they cannot get help because they cannot reach the hospital, and they are just dying at home,” Mr. Zhovnyakh said. “What is happening now in Ukraine is a humanitarian catastrophe caused by the war. The world is watching us, praying for us, and not doing much else. Ukraine, unfortunately, is on its own.”

The breakdown in Kyiv’s public-transport system—the metro is now used as a bomb shelter—means doctors and other staff have a hard time getting to work. They are now arranging to travel together in shared cars, and often sleep in the hospital’s basement. Volunteers bring in water and food. On Tuesday morning, a local franchise of Domino’s delivered a van full of pizzas for patients and staff.

Asked by a surgeon about her mood on Tuesday, one of the nurses raised her fist in a greeting of Spanish antifascists during the siege of Madrid in 1936 and said, “No Pasaran!,” or “They shall not pass!” Madrid eventually fell in 1939.

In the basement shelter of the Okhmatdyt hospital, Valentyn Vetrov has been staying since Thursday with his 1-year-old son, Ilya, who has had more than 20 surgeries to fix birth defects. His eldest son is fighting in the Ukrainian army, he said. His wife and five other children have remained behind in the Azov Sea town of Berdyansk, which is now occupied by Russian troops.

“If not for the baby, I would be on the front lines too, fighting until my last breath,” he said. “But for now my duty is here.”

Sharing the basement with him was Ludmyla Kmetyuk, an entrepreneur from the western Ukrainian town of Khmelnytsky. Her 8-year-old son, Yaroslav, just had surgery for a brain tumor and needs physical therapy and other specialized care to be able to move on his own. That is impossible now, and Yaroslav lays listless on a blanket in the basement.

“What a shock. I never imagined that they would do this to us,” Ms. Kmetyuk said. “Why? What have we done to them?”