As the volunteers unpack their shopping bags of rice, milk and cooking oil, Goman Badder retreats to the room he lives in with his wife and their one-year-old son. For the 28-year-old Syrian Kurd, the deliveries are a mixed blessing: he is relieved that his family will not go hungry, but humiliated that it has come to this. He had hoped for better things for his son, asleep on the neatly made bed.

"When I left Syria, I felt I didn't want him to be like me," he says. "But I never thought for him it would be like this: people bringing him Pampers and milk. If I'd known it would be like this I would never have brought him to this world."

His friend Salah Muhamed, a Kurdish teacher who fled the "hell" of Syria six months ago, did not mince his words. "In Syria, we will be killed by guns," he says. "Here, we will be killed by the economy."

For the past six months, this state-funded centre for asylum seekers has received no funding from the state. Its 25 staff members have not been paid since January. Due to its debts, the centre has "huge problems" with suppliers and the usual food deliveries stopped two weeks ago.

Some of the 225 residents thought the staff had gone on strike over pay but, as director Vasilis Lyritzis explains, they could simply not go on: "We just stopped cooking because we didn't have anything to cook. The moment that we have food, we cook again."

Even now, the inhabitants of the Lavrio centre, about 40 miles from Athens, are among the most fortunate of Greece's tens of thousands of asylum seekers, most of whom receive no support from the state and sleep rough in large, unsanitary slums in cities such as Athens.

But, as the debt crisis takes its toll on state services and the unpaid bills to suppliers mount up, Badder and Muhamed are being pushed to the extremes. And they are not the only ones. Across the country, some of Greece's most marginalised and vulnerable people who rely directly or indirectly on the state to feed them are now facing the possibility of having to turn to their neighbours for help.

On the island of Leros in the Dodeccanese, the governor of a psychiatric hospital has, in the past two weeks, like Lyritzis, found himself having to plead with suppliers to keep bringing food, despite an almost complete inability to pay them. Yiannis Antartis's suppliers stopped for a week, after which he found enough money to pay them each €15,000 (£12,000) – enough to encourage them to restart but far from enough to cover the hospital's total debts to them of €1.25m.

Antartis believes he has about a month to find money to pay the debts and protect the 400 patients, who have a range of mental illnesses, from depression to dementia and schizophrenia. Owed about €13m by its debt-ridden health insurance funds, the hospital, which is already sending samples to private labs for want of sufficient facilities, will soon be struggling for the basics, he said.

"If we don't receive money within three to five weeks we're going to have a real problem – for food, for medicines, for hospital supplies, bandages, even the basics," he says. "All levels of the hospital are going to start breaking down.

"My primary concern is to receive enough money to pay some outstanding bills in order to keep the hospital going, from the insurance funds, from the state, from wherever, just to be able to keep the hospital going. I am really worried about the fact that the hospital will not be able to function normally if we don't receive payment."

His position will be met with sympathy by the director of a state hospice in the Kypseli neighbourhood of Athens, who said last week that, as well as medicine shortages and an inability to pay energy bills, the institution had had no meat or chicken since 1 June.

Antartis has been the hospital's governor on and off since the 1980s, when it was revealed to be keeping patients naked and chained to their beds. He is adamant the institution bears no semblance to its former self. In 1988, he says, 1,600 patients were "piled up" in wards and looked after by fewer staff than today. Now, 120 of the 400 patients live relatively normal lives outside the hospital.

But, last week, Antartis was so concerned about supplies that he wrote to the health ministry and major political parties warning that patients were being poorly fed. Since then, he has managed to get the supplies coming in again and has met Greece's caretaker health minister, who, he says, showed understanding and promised a €150,000 payment by the end of June.

"But he [the minister] emphasised that the insurance funds do not have any money and that the hospitals have to keep every penny in order to keep going," Antartis says. Given the size of the debts and the interest rates on them, €150,000 "doesn't cover anything".

The inability of health insurance funds to pay their debts has caused chaos in the healthcare system, contributing to a shortage of medicines and the closure of dozens of commercial pharmacies.

Antartis is relieved the hospital managed to resolve – albeit temporarily – the problem with food supplies without the patients realising anything was wrong. But he remains concerned, particularly about medical supplies, as that is one area in which Leros's community, as active and kind as it is, will not be able to help. "I think the local people will help with food," he says. "But even that is going to be a problem at some point because for how long are they going to be able to help?"

One place where Greeks have reportedly already stepped in to fill the gap left by the state is Corinth, where the prison has had food shortages and people have started collecting goods for the inmates. Another is Lavrio, where donations have been keeping food on the table for the asylum seekers and refugees, among whom Lyritzis estimates there are 75-80 children. Some of the volunteers are hardly well-off themselves: Nadia, one of the members of a people's assembly from Athens which dropped off food at the weekend, is on Greece's new unemployment benefit of €360 a month and has had to go back to live with her mother.

"€40 or €50 just means I won't go out for a couple of days," she shrugs, loading bags of groceries into a waiting car.

Lyritzis, an employee of the Hellenic Red Cross which runs the centre, says the finance ministry has been unable to tell him when, or if, its funding might come. He says the centre had an estimated five days of food supplies left and he is in a very bad psychological state.

Contacted by the Guardian, the ministry of health says the centre's funding has "not been axed in any way" and that payments have been delayed due to a legal obstacle raised by the financial audit office. A spokeswoman says it has been following up the issue on a daily basis. "We hope to resolve this issue in the next few days," she says.

For Badder, who came to Greece in 2010 after alleged persecution due to his Kurdish ethnicity, that would be good news.

But if the situation is not alleviated, he would like to see the responsibility for his centre fall to a power outside Greece. He says: "The Greek people are very, very good. They look after us. But the Greeks have no money. Why must the Greeks bring us food? We are in Europe, not just Greece."

As the volunteers busied themselves in the courtyard, unpacking tins, toiletries and cartons of milk, he says: "The people here are so nice. We can't tell them: 'Don't bring food because we are ashamed'. We must say 'Thank you'. But we feel bad."

• This article was amended on 14 June 2012. The original was garbled during the editing process, losing a paragraph including quotes from the Greek ministry of health.