WASHINGTON – Starting Tuesday, an estimated 50 million Americans who do not have health insurance are to start signing up for coverage as the core of President Barack Obama’s overhaul takes hold. Subsidies will be available to help those with lower incomes pay for insurance. And on Jan. 1, people will no longer be denied coverage because of previous illnesses.

Yet Americans are hardly celebrating.

Polls show most of them do not like the program. Many object to requirements that all Americans have health insurance or face fines.

Some employers, the main provider of health insurance in the U.S., are citing the program as they cut back on coverage — contrary to Obama’s promise that people could keep their coverage without worrying about anything changing.

Republicans have waged an unrelenting battle to repeal the law, claiming the program will cost jobs. They have had no success so far, with Democrats controlling the Senate and White House.

But even without Republican actions, “Obamacare,” as the program is commonly known, is already off to a rocky start. Obama has had to delay a key part for one year: a mandate that large employers provide health care coverage. On Thursday, the government announced delays for small businesses seeking to buy insurance online.

“This law is a mess. It needs to go,” the Senate’s top Republican, Mitch McConnell, said in a speech Thursday. “It’s way past time to start over.”

In the run-up to Tuesday’s deadline, Obama has been trying to boost support for the plan. In a speech Thursday just outside Washington, he rejected Republican claims that it will further damage an economy that still has not fully recovered from a lengthy recession.

“The evidence is that it’s not going to hurt the economy,” he said. “Obamacare is going to help the economy. And it’s going to help families and business.”

The success of the health care overhaul, Obama’s major legislative achievement, could largely define his legacy. Obama won its passage early in his presidency while Democrats controlled both chambers of Congress. States challenged its constitutionality, but the U.S. Supreme Court narrowly upheld the core of the program.

The United States has long been the only major developed nation without a national health care system. Even the Affordable Care Act, as the program is formally called, will not have a single, government-run health insurance program covering all Americans.

Instead, the program relies on health insurance marketplaces in every state to provide options for people who do not have job-based coverage, who cannot afford an individual policy or have a health problem that disqualifies them under the current system.

The federal government will run the markets in the primarily Republican-controlled states that have refused to do so. The coverage will not be free, even after sliding-scale subsidies keyed to income. But Medicaid, the federal-state health insurance program for the poor, will be expanded in states that wish to participate.

The combination of subsidized private insurance through the new markets, plus expanded Medicaid could reduce the number of uninsured by half or more in several years, government budget agencies say.

Under the current system, most Americans have health insurance as a benefit of employment, with the employer and the employee sharing costs. The federal government covers the elderly, and many of the poor are covered by Medicaid.

Some people who don’t have health insurance through work buy individual coverage — often policies that just kick in in the case of medical catastrophes. Young people often forgo the expense of insurance, relying on their generally good health. Those without health insurance, especially the poor, often seek treatment at hospital emergency rooms, which are banned by law from denying care to patients who cannot pay.

The reliance on hospital emergency rooms by those who are sick but uninsured helps drive up medical costs, which are higher in the United States than in other developed countries. Hospitals increase their charges to recoup the expense of treating the uninsured, passing on the cost to those with insurance.

Also, insurance companies can refuse to insure people who have medical problems and can set payment limits for those who have insurance but exceed a set amount in claims, which is particularly difficult for those who develop a severe and expensive medical problem.

The Affordable Care Act forbids insurance companies from those practices.

The law is going into effect in stages and has already forced insurers to allow children to stay on their parents’ policies through age 26. It has also set a limit on the percentage of insurance company income that can go to profit. That has seen some companies issue refunds to their policy holders. More changes will take effect in the coming years, such as a new tax on large employers that offer workers generous plans for which the higher premiums are tax-deductible.

Consumers have until Dec. 15 to sign up for coverage starting Jan. 1. The grace period runs through Mar. 31 to avoid penalties for 2014. Fines start as low as $95 the first year but escalate thereafter.

Much of the opposition to Obamacare stems from that requirement to buy insurance. Many Americans view this as too intrusive. Republicans, particularly those allied with the small-government tea party movement, have made repealing the program their core issue, although the idea on which it is based was historically promoted by Republicans.

Republicans also say that the plan will kill full-time jobs. They point to smaller employers that are moving workers to part-time status or shedding them to stay below the 50-employee minimum that will force them to comply with providing health insurance benefits. Larger organizations, such as fast-food chains, also are cutting hours to keep employees at less than full-time status, since that would guarantee minimum health coverage.

Polls give Republicans some encouragement to press on. A Pew survey shows just 42 percent of those questioned back the new law, with 53 percent opposed. However, those who disapprove are divided over what lawmakers should do next. The poll found about half of the disapprovers say “lawmakers should do what they can to make the law work as well as possible.”

The Republican-led House of Representatives has voted more than 40 times to repeal or delay the law, only to see the legislation die in the Democratic-led Senate.

Obama casts the Republican opposition as purely political.

“The fact is, the Republicans’ biggest fear at this point is not that the Affordable Care Act will fail,” he said. “What they’re worried about is it’s going to succeed.”