Some 29% of people had no leftover cash to put into an account which could cover emergency bills, StepChange Debt Charity found.
A further 19% said that even when they could save, they had £50 or less available each month.
Nearly half (45%) of people earning less than £20,000 had been unable to save in any of the previous 12 months, according to the charity's survey of more than 3,200 people.
StepChange said many households are living in a "perilous financial state", with a fifth saying they would not be able to survive financially for more than a month if their income fell by a quarter.
The charity said the scheme to automatically enrol people into workplace pensions should be adapted so it also encourages people to build up a precautionary savings pot.
Households are expected to face tougher economic conditions this year. Office for National Statistics figures recently showed the Consumer Price Index measure of inflation reached 1.6% in December, marking the highest level since July 2014.
Bank of England figures have recently shown strong increases in consumer credit, prompting concerns that some people may be over-stretching their borrowing.
Personal insolvencies jumped by 13% year-on-year across England and Wales during 2016, according to figures released by the Insolvency Service last week.
Mike O'Connor, chief executive of StepChange Debt Charity, said: "Not having a precautionary savings pot means that people are extremely vulnerable to falling into severe problem debt.
"For too many people stretched budgets leave no room to put even a little aside as they struggle to get by, just about managing month after month.
"A drop in income or an unexpected expense pushes people over a cliff from managing to crisis."
A Government spokesman said: "This Government is committed to supporting savers at all income levels.
"We are helping people to save more through the rollout of automatic enrolment and introducing Help to Save, giving everyone the opportunity to plan ahead."