Not all who need health insurance are happy with their new options. Many have complained that the prices are too high, especially if they earn too much to qualify for federal subsidies. And many will have a limited choice of doctors and hospitals under the new exchange plans, which have “narrow networks” to hold down premiums.

But Mr. Acosta, like the people in the following profiles, says the health care law has given him a cautious sense of hope.

He lives alone and has turned to his siblings at times for help with his insurance and medical costs. And those costs have at times been high: He sees an oncologist in Tucson twice a year and sometimes needs expensive tests that his current insurance does not always cover. He has also recently been stretching his medication, a drug called Gleevec, which helps make his type of leukemia manageable, to ensure he would have some on hand if he became uninsured.

“After being gouged all these years, trying to make ends meet, to all of the sudden get this?” he said. “I’m really blown away.”