In the beginning, he said, “always I thought, how long before I go out? Because when you wake up in the same room every day it’s the same thing, ‘When I can get out?’ It’s always depressing. But day by day, day by day, you don’t need to worry about what will happen, because when you wake up it’s always the same room.”

RAYMOND FOK was born in Madagascar and grew up in Hong Kong, where he became a police officer. In 1988, he brought his wife and two young sons on what he told officials was a vacation in New York, and then never returned. Mr. Fok left some question about his reasons for overstaying the family’s tourist visas, repeating that he had feared Hong Kong’s approaching handover to the Chinese government, though at the time this was nine years away.

In New York he found a job at a vegetable market in Chinatown, earning $5 an hour to feed a family of four — and soon, with the birth of his daughter 18 years ago, five. A friend helped him rent an apartment in a heavily Chinese section of Bensonhurst, Brooklyn, and taught him to navigate the subways. But the friend refused to help him apply for permanent residence, Mr. Fok said.

Eventually he landed a job driving a truck for a Chinese-owned company in New Jersey, at $400 a week, off the books, with no insurance benefits, he said. He had a driver’s license and bought a car to commute.

“Make a life, pay rent, support a family,” he said. His wife worked in a laundry on Delancey Street. His sons went to school and later found jobs in bodegas or bagel shops. It was enough.

But driving was stressful, with no extra pay for overtime, and he lived on fatty foods consumed on the go. When his kidneys failed, an emergency-care provision in Medicaid paid for dialysis treatments, though he was otherwise ineligible for coverage.

For Mr. Menkes at New York Downtown, any day the emergency room door might let in the next Mr. Fok. Under federal law, hospitals are required to treat anyone who comes in, regardless of his or her immigration status or ability to pay. Half of the in-patients at Downtown are Asian immigrants, many of them undocumented, Mr. Menkes said. Forty-five percent of those on the staff speak at least one dialect of Chinese. When Mr. Fok arrived — with no Social Security number, no green card, no insurance, disoriented, no known family or address — the hospital was ready for him.