Six hospitals in the Indian capital Delhi have completely run out of oxygen and doctors say other hospitals have just a few hours' worth of supply left.
A number of people have died while waiting for oxygen, and more than 99% of all intensive care beds are full.
India is in the grips of a second wave of Covid infections and on Thursday recorded the highest one-day tally of new cases anywhere in the world.
The country has recorded close to 16 million confirmed infections.
There have been 314,835 new coronavirus cases in the past 24 hours, while deaths rose by 2,104.
There have been reports of state authorities stopping oxygen tankers from travelling to other states, according to Delhi television station NDTV. Some facilities have been accused of hoarding their supplies.
Indian politician, Saurabh Bharadwaj, who is being treated in a Delhi hospital for Covid, posted a plea for help in Hindi on Twitter, saying there were just three hours of oxygen left where he was.
"A lot of people are dependent on oxygen and without oxygen, these people will die just like fish die in the absence of water," he said. "This is a time for all to come together to work."
Delhi is known to have among the best healthcare facilities in India, but it has been brought to its knees by the latest surge in cases, says BBC India correspondent Yogita Limaye.
Families are also waiting hours to perform funeral rites, Reuters news agency reports, with at least one Delhi crematorium resorting to building pyres in its car park in order to cope with the numbers arriving. Crematoriums are holding mass cremations, and working day and night in several cities.
"During the first phase of coronavirus, the average here was eight to 10, one day it reached 18. But today the situation is very bad. Last night we cremated 78 bodies," Jitender Singh Shunty, who runs a crematorium in northeast Delhi, told Reuters.
"It is four times more frightful, this coronavirus... Many bodies are around, waiting. We have no place left in the crematorium to cremate them. Very bad times, very bad times," he added.
A doctor working in a government hospital in the south of India, who wished to remain anonymous, said tensions were running high.
"Patients are trying to hit doctors," they told the BBC. "They are blaming doctors for everything and even the [hospital] management is also blaming the doctors. It's a stressful environment."
"We have presently almost used 99% of oxygen ports - only 1% is left. It's a very pathetic situation."
One family's story
Sukhbinder Singh lives in Leicester in the UK and told the BBC that his mother, 90-year-old Harbans Kaur, died on Wednesday in Delhi after she and seven members of his family tested positive for Covid-19.
A nursing home refused to admit Ms Kaur without a family member to care for her, but no-one was well enough to do so and she had to stay at home.
"After several phone calls and pleading, I managed to organised the cremation through a charity for my mother, which was carried out with some usual rituals," he added. "But there were no final farewells by family and the memory of her death is very difficult to deal with mentally."
Mr Singh's brother and sister-in-law were admitted to hospital in Delhi, and using his contacts in India, he was finally able to secure the medicine and oxygen they needed - although he said he struggled in part because of rising medicine costs due to the black market.
"My lower-middle-class family is not rich to afford big private hospital fees and treatment costs. What about poor daily wagers who can't even afford paracetamol?"
In January, the pandemic was relatively under control, but since then the situation has gone from bad to worse, for a number of reasons.
Political parties have come under criticism for holding huge state election rallies - including one by Prime Minister Narendra Modi - in West Bengal. The government has defended the decision to continue with polling, which is taking place in phases, but India's election body has now banned campaign rallies.