Their homes destroyed by waves or contaminated by radiation, some 150,000 Japanese now live in places where they used to wait for trains, attend school or play pickup basketball.

Relief workers worry about unsanitary conditions and mounting stress. Local government officials worry that prefabricated houses aren’t arriving fast enough. Evacuees worry about the months to come, and the dubiousness of searching for new homes as permanent as the ones they lost.

In the month since an epic earthquake and tsunami destroyed Japan’s northeastern coast, evacuation centers have transformed from places where people can stay into places people wish to leave. Most shelters now feature heat and running water and three good meals a day. But they don’t offer privacy, a place to grieve or an opportunity to rebuild. The problem is, few evacuees have found other places to go.

On Monday, Japan’s government announced that it would enlarge the evacuation radius around the crippled Fukushima Daiichi nuclear plant beyond the current 12-mile limit, triggering another influx of evacuees.

Though government spokesman Yukio Edano said the risks of a widespread a radiation leak had become “considerably lower,” he said that certain areas within 12 to 19 miles of the plant, based on radiation readings, will be given mandatory evacuation orders.

Even as Japanese authorities struggle to cope with the forced migration, scrambling to move evacuees into hotels or apartments where conditions are more home-like than in the mass original evacuation centers, they predict that many shelters will continue to operate for months.

It’s a sign of this catastrophe’s magnitude that even those leaving the shelters are moving only to slightly less temporary housing, often far from home. In towns where most buildings have been flattened, there’s still no good answer for the looming question: Where will people go?

“I think it is only when the temporary housing is dismantled and removed that we will be heaving a sigh of relief, when we feel that things are back on track,” said Yasuhiro Kanno, a local government reconstruction official in Rikuzentakata. “But it will take a long time.”

According to statistics from Japan’s national police agency, the March 11 disaster collapsed 47,776 buildings. Another 11,030 partially collapsed. Some 84 burned down. Another 2,736 were flooded above floor level.

In coastal Rikuzentakata, the damage was particularly fierce, with a tsunami wave grinding the town into a de facto landfill. But this weekend, Rikuzentakata become the first town in Japan to offers its evacuees prefab housing.

Thirty-six units were constructed. Each measured roughly 320 square feet. The Japanese Red Cross furnished each with a rice cooker, a microwave and a television. The homes — grayish trailers with wooden panels — were assembled on the grounds of a middle school. Normally, they retail for $23,000.

By the time Rikuzentakata officials were ready to give away the keys, they had 1,160 applicants for three dozen trailers. They put applicants’ names in two boxes, using one box solely for candidates who are elderly, handicapped or have small children. Town officials picked 18 names from that priority box, 18 names from another.

They called the winners. They didn’t notify the others.

Officials in Rikuzentakata say their area ultimately will need 3,600 prefab homes. They hope to receive the allotment by August.

Combined, Japan’s hardest-hit prefectures — Miyagi, Iwate and Fukushima — have requested 60,000 prefab homes. Japan’s prefab construction association said it can build 30,000 within the next two months; the other 30,000 within the following three months. But for now, the construction association faces a problem: Local authorities still don’t know where to put the new buildings. So far, land for 8,000 has been secured.

“If land is secured we can start work right away,” said Hironobu Sato, an official at the Japan Prefabricated Construction Suppliers and Manufacturers Association.

Residents will be asked to stay in prefabricated homes for no longer than two years. In the meantime, those who wait in shelters have drawn increased attention from relief workers, who say that sanitary conditions still require improvement.

John Sparrow, an International Red Cross spokesman, said that at one Fukushima evacuation center, a blocked toilet has forced residents to defecate on paper, then wrap the paper and dispose of it in a plastic bag.

Towns are left with a difficult calculus, particularly as schools — a common shelter site — attempt to reopen for students. Many local governments, according to aid workers, are making attempts to break apart their largest evacuation sites, as conditions in arenas and conference centers tend to offer the least privacy. Other towns have erected makeshift barricades, allowing personal space for families.

The luckiest evacuees get to leave the shelters entirely.

Last weekend, 360 evacuees from Fukushima arrived at the Grand Prince Hotel Asakasa, marble-lobbied 40-story building in downtown Tokyo that once fed the capital’s most ostentatious appetites.

Now, the building is closed, worn down and due for demolition. But Tokyo’s metropolitan government approved a $2.4 million plan that allows evacuees to stay there through the summer, meals included — all for free.

Monday morning, 55 evacuees attended a welcome meeting in a conference room, spreading out on plush pink chairs. A mother sat with her daughter. Husbands sat with wives. An old man sat alone. All wore green lanyards with ID cards.

Four ward officials, in a 15-minute presentation, explained the nuances of the evacuees’ new neighborhood: the location of the hospitals, the grocery stores and the subway stops. They gave the hours at which a lawyer would be on call for concerns about inheritance and insurance. They also explained a guarantor-free loan being offered.

“As for your kids,” said Hideaki Hirai, the Chiyoda ward education official, “they will be transferring tomorrow. We will pay all expenses for books, pencils and so on, as well as the school-provided lunch. Also, in the morning, there will be a bus that takes them to school. But you will have to pick your kids up in the afternoon.”

Special correspondents Akiko Yamamoto and Tetsuya Kato contributed to this report.