ZARASZÓW, Poland— Krystyna Trzcińska, 68 years old, has farmed a strip of land in this corner of eastern Poland for more than four decades. Retired now, she grows clover between neat rows of raspberry bushes to feed her rabbits. The rabbits she eats, the berries she sells.

The berries bring in the equivalent of about $1,300 a year. To survive, she and her husband depend on pensions provided by Poland’s government.

State-funded pensions are at the heart of Europe’s social-welfare model, insulating people from extreme poverty in old age. Most European countries have set aside almost nothing to pay these benefits, simply funding them each year out of tax revenue. Now, European countries face a demographic tsunami, in the form of a growing mismatch between low birthrates and high longevity, for which few are prepared.

Europe’s population of pensioners, already the largest in the world, continues to grow. Looking at Europeans 65 or older who aren’t working, there are 42 for every 100 workers, and this will rise to 65 per 100 by 2060, the European Union’s data agency says. By comparison, the U.S. has 24 nonworking people 65 or over per 100 workers, says the Bureau of Labor Statistics, which doesn’t have a projection for 2060.

While the problem has long been building, it is gaining urgency as European countries’ debt troubles, growing out of the 2008 crisis, push governments to reassess their priorities. Greece, the worst off, has had to reduce the generosity of its pension system repeatedly. Though its situation is unusually dire, Greece isn’t the only European government being forced to acknowledge it has made pension promises it can ill afford.

“Western European governments are close to bankruptcy because of the pension time bomb,” said Roy Stockell, head of asset management at Ernst & Young. “We have so many baby boomers moving into retirement [with] the expectation that the government will provide.”

Even the U.S., with a Social Security trust fund of $2.8 trillion, faces criticism for promising more than it can afford. That is because the fund—which is mostly in the form of IOUs from the Treasury—is projected to fall short of the sums needed to cover all benefits in a dozen years or so, and run out in 2035. Europe’s situation is much worse.

The demographic squeeze could be eased by the influx of more than a million migrants in the past year. If many of them eventually join the working population, the result could be increased tax revenue to keep the pension model afloat. Before migrants are even given the right to work, however, they require housing, food, education and medical treatment. Their arrival will have effects on public finances that officials have only started to assess.

The pension squeeze doesn’t follow the familiar battle lines of the eurozone crisis, which pits Europe’s more prosperous north against a higher-spending, deeply indebted south. Some of the governments facing the toughest demographic challenges, such as Austria and Slovenia, have been among those most critical of Greece.

Germans, meanwhile, “are promoting fiscal rules in Spain and other countries, but we are softening the pension rules” at home, said Christoph Müller, a German academic who advises the EU on pension statistics. He pointed to a recent change allowing some workers to collect benefits two years early, at 63. A German labor ministry spokesman called that “a very limited measure.”

Europe’s state pension plans are rife with special provisions. In Germany, employees of the government make no pension contributions. In the U.K., pensioners get an extra winter payment for heating. In France, manual laborers or those who work night shifts, such as bakers, can start their benefits early without penalty.

While a few countries—including Norway, the U.K. and the Netherlands—have considerable savings in public funds or employer-sponsored pension plans, many others have little. Governments’ annual costs for public pensions equal a tenth of gross domestic product, according to the EU data agency Eurostat. That GDP percentage should be stable in coming decades, Eurostat estimates, though its forecast depends on numerous economic assumptions.

Across Europe, the birthrate has fallen 40% since the 1960s to around 1.5 children per woman, according to the United Nations. In that time, life expectancies have risen to roughly 80 from 69.

In Poland, birthrates are even lower, and here the demographic disconnect is compounded by emigration. Taking advantage of the EU’s freedom of movement, many Polish youth of working age flock to the West, especially London, in search of higher pay. A paper published by the country’s central bank forecasts that by 2030, a quarter of Polish women and a fifth of Polish men will be 70 or older.

In 2012, the Polish government launched a series of changes in its main national pension plan to make it more affordable. One was a gradual rise in the age to receive benefits. It will reach 67 by 2040, marking an increase of 12 years for women and seven for men. The changes mean the main pension plan now is financially sustainable, said Jacek Rostowski, a former finance minister and architect of the overhaul.

The party that enacted the changes lost an election in October, however, and a central promise of the winning party is to undo them. Recently, Poland’s president introduced a bill to reverse some of the measures.

“You have to take care of people, of their dignity, not finances,” said Krzysztof Jurgiel, agriculture minister in the current Law & Justice Party government.

Ms. Trzcińska, the retired grower of berries and rabbits, doesn’t follow politics too closely. She switches channels when political debates such as the one over pensions appear on her TV screen. “They are all yelling at each other, I don’t understand it, and it’s unpleasant,” she said.