On the day in 2010 that he put his signature to the biggest domestic achievement of his presidency, President Obama declared: “After a century of striving, after a year of debate, after a historic vote, health-care reform is no longer an unmet promise. It is the law of the land.”

Now, we get to see whether it works.

Starting Oct. 1, more than 40 million Americans who lack medical insurance will have their first chance to sign up for coverage under the new system that has come to be known as Obamacare. That sets the gears in motion to begin what the president has promised will be a transformation of the nation’s entire health system — making it fairer, less costly and more effective at providing care.

But while Obama might have thought that signing the new law would begin a political healing process, the divisions three years later are as deep as ever.

Poll after poll has shown Americans remain skeptical and confused about Obamacare. And on the day enrollment opens, the federal government may shut down as part of a budget battle tied to a Republican effort to block the law’s implementation. But the law will move forward, in any case.

Having passed a test of its constitutionality before the Supreme Court, the law now faces an even more critical judgment — one that will be written in millions upon millions of individual stories.

Among the winners will be the uninsured, assuming the coverage they get proves to be affordable and adequate. Businesses facing a new requirement to provide insurance say they could be losers. Some people who buy their own insurance may pay more, while others may pay less. Hospitals and doctors welcome the fact that more patients will be able to pay their bills but worry the government will demand more say in what kind of care they provide.

“The law is entering a new phase where it becomes real,” Drew Altman, president of the Kaiser Family Foundation, said. “A lot is going to turn on how smooth the process is, whether people get coverage and whether they like the coverage they get.”

In coming months, clues will emerge on whether the system will succeed or fail: Will the new insurance marketplaces work as designed? Will enough Americans — especially young, healthy ones — sign up? Will patients be able to find enough primary-care doctors and specialists — and see the ones they trust?

“We’re now seeing that a lot of the exchanges probably will have very limited networks and very low reimbursements to providers” compared with the large employers’ plans, said Douglas Holtz-Eakin, a former head of the Congressional Budget Office who now heads the conservative American Action Forum think tank. “This is the Medicaid story: Sure, I’ve got insurance, but no one wants to see me, and so I go to the emergency room for regular care.”

Towson resident Aniela Russo will start a new job next month and likely make too much to stay on Medicaid, the state-federal health program for the poor. So she’ll have to turn to the health marketplace for coverage. “I’m going to be devastated if I can’t keep my doctor,” said Russo, who suffers from a genetic heart condition. “It really feels like a matter of life or death.”

Most Americans will continue to get coverage through work; beginning in 2015, companies with more than 50 people will need to cover employees who work at least 30 hours a week.

Many of America’s poorest uninsured won’t be helped by the law. Currently, many low-income people, including childless adults, are ineligible for Medicaid in many states. The health law sought to change that by expanding Medicaid to include anyone with an income at or below 138 percent of the federal poverty level ($15,400 for an individual). However, the Supreme Court ruled that states could opt out of the expansion and about half have done so. Under the law, subsidized insurance on the marketplaces won’t be available to those below the federal poverty level ($11,490).

“We’re going to have an upside-down social program in these states” that do not expand Medicaid, said Massachusetts Institute of Technology economics professor Jonathan Gruber, who advised the administration on the law. “If you’re poor, you get nothing. If you cross the poverty line, you can get subsidies and buy into the exchanges.”

In Prince George’s County and Southern Maryland, Greater Baden Medical Services estimates that 40 percent of the patients at its community health facilities are uninsured. It will be working feverishly to help them sign up for coverage. But providers there have no idea how many more patients to expect, once the law kicks in Jan. 1.

“Nothing this big is going to happen without glitches,” said Sarah Leonard, who is about to retire as Greater Baden’s chief executive. “We know there’s a long line of people needing care. But how soon will they enroll? What will they expect or need? What other providers will there be for them? Those are some of the great unknowns.”

The health law’s success rests on Americans buying coverage en masse. Low enrollment could cause premiums to spike. But some young people already are planning on paying the penalty, rather than buying coverage by Jan. 1, as required.

A 26-year-old Arlington lawyer, who has $200,000 in student loans and works on contract for firms, figures his fine would be $600. That’s far less than the almost $1,700 a year he would pay for an inexpensive plan. “Money is tight and the income isn’t consistent,” he said, speaking anonymously so as not to hurt his search for a permanent job.

In fact, premiums have, so far, come in below government forecasts. But because of new rules, some young, healthy people with low rates may pay more than they do now, while some older, ailing consumers might pay less. And premiums are only part of the story. Some low-premium plans have high co-pays and deductibles.

Experts say it hasn’t been clear how consumers will react.

Now, America is about to learn.