Now, the two-dozen remaining exiles are aging at a pace that is outstripping the Navy’s ability to care for them. The Navy flies the seriously ill to military hospitals in the U.S. Navy personnel have converted former nurses’ quarters at the base hospital into a 24-hour assisted-living facility. Government drivers transport the Cuban exiles to doctor appointments, exercise classes, McDonald’s and the all-in-one base store, the Navy Exchange.

“At the time, the Navy offered them safe haven, and we said, ‘Hey, you’re welcome to stay here until this gets resolved,’ ” said the base commander, Navy Capt. John Nettleton. “And here we are half a century later, and they’re still here.”

With few exceptions, they have never returned home. Many have made their way to the U.S. But a core group chose to stay, even though they acquired U.S. citizenship or residency. The U.S. Navy provides them free housing, utilities and medical care, along with subsidized meals at base mess halls.

“I thought I was only going to be here for six months,” said Mr. Baudin’s neighbor Noel West, 81 years old.

The ill are treated at the Navy hospital, and the dead buried by the beach in the base cemetery, alongside sailors and Marines who perished in the tropics 100 years ago.

For more than 50 years, the exiles have waited out Mr. Castro, circumscribed by a 17-mile razor-wire fence that separates their present from their past. They have married and divorced, had jobs and children. They have danced at base clubs and drunk at base bars. They play dominoes and listen to singer Celia Cruz. They have also seen their adopted home become synonymous with prisoner abuse since the U.S. housed nearly 800 terror suspects here .

Mr. Baudin has been here ever since, part of a small group of Cuban exiles who, in a hot moment of the Cold War, won permission from the U.S. government to stay at the Guantanamo Bay Naval base until Cuba was free.

NAVAL STATION GUANTANAMO BAY, Cuba—In 1961, Ramon Baudin got wind that Fidel Castro’s security forces were looking for him. He hid in a bus headed to this U.S. military base, sneaked past a police checkpoint, then pleaded with the American sentry: “Hey, buddy, I’m running away. Open the gate.”

U.S. troops landed at Guantanamo Bay during the Spanish-American War of 1898, and in their victory over Spain signed a perpetual lease that can only be canceled by mutual agreement between Havana and Washington.

Thousands of Cubans once commuted to base jobs. But relations with the U.S. soured quickly after Mr. Castro seized power in 1959, with the countries severing diplomatic ties in 1961.

The Guantanamo Cubans have been buffeted by the long conflict between the U.S. and Cuba, as well as the personal decisions they made long ago. When Washington and Havana were inches from war in the 1960s, the exiles found shelter on the base but gave up their homes and families.

Now that President Barack Obama has decided to normalize relations with Cuba, their fate is again up in the air. The Navy is unsure whether the easing of tensions means the exiles will have to move out.

For now, the Guantanamo Cubans are staying put, accustomed to the narrow but easy life on the military base, and skeptical that warmer ties with the U.S. means the Cuban regime would treat them well if they returned home.

“It’s kind of early,” said exile Ramon Romero, who took refuge on the base in 1960, at age 17. “You can never trust Castro because he can say one thing and turn around and do another thing.”

At 92, Mr. Baudin is one of the oldest of the 23 Cubans—known officially as special category residents—remaining on the base.

He says he isn’t homesick. But every day after the sun comes up, he settles into a white plastic chair under the roof of his carport and switches on a portable radio. Despite a broken antenna, it picks up a station from Caimanera, a town just outside of the barbed wire. Between news broadcasts and fast-paced guarachas, he listens for obituaries of long-lost friends.

“I can’t go there anymore,” he said of Caimanera, drawing a finger across his throat. “They’d kill me.”

A Cuban government spokesman in Washington didn’t return calls or emails seeking comment on whether the exiles would face legal or political peril if they returned home.

Mr. Baudin took refuge on the base when Cold War tension was aggravated by the botched 1961 Bay of Pigs invasion and the Cuban missile crisis a year later.

Mr. Baudin was engaged to a woman in Caimanera when he learned that regime security men were after him. He never had a chance to explain his disappearance. He heard she married a milkman.

Carmen Gonzalez fled to the U.S. Naval base on Guantanamo Bay with her mother in 1964. She married, had a daughter, became a widow and grew old all while living within the confines of the 45-square-mile military facility. Photo: Brandon Thibodeaux for The Wall Street Journal

For work, Mr. Baudin was a caddie on the base golf course, then spent decades pressing uniforms in the laundry. He remembers good times with his Cuban friends here, fishing in the bay and drinking rum on the beach.

Most of them are dead now. Mr. Baudin wakes at about 3:30 each morning and meditates. Then, before dawn, he walks the streets, talking, he says, with the ghosts of his fellow exiles: He sees Gustaff Polica, who raised two children on Guantanamo and whose body is buried in the hills. He spots a woman in white who went home to Cuba to die with her family after decades at Guantanamo.

“I pray for my dead friends,” Mr. Baudin said. “They’re by my side. They watch over me.”

Most of the Cubans live in a neighborhood of flat-roofed concrete houses built in 1960, their pastel blues and greens faded by the sun. From his window across the way, Mr. Romero often watches Mr. Baudin pass under a streetlight on his predawn walk.

As a boy, Mr. Romero accompanied his grandfather on a 30-foot banana boat, steering it along the shore. They stopped at the base each day to sell fruits and vegetables.

By 1960, the family’s standing with the Castro government had become precarious. Two of Mr. Romero’s cousins had been killed, and another was languishing in prison, he said.

A home in the neighborhood where most of the 23 remaining Cuban exiles live. Photo: Brandon Thibodeaux for The Wall Street Journal

Mr. Romero’s grandfather, who cared for the boy, sent Mr. Romero to live on the base. “My grandfather said, ‘Stay here, otherwise you’ll be in jail or dead,’ ” Mr. Romero, now 71, recalled. “So I’ve been here ever since.”

Mr. Romero at first cried himself to sleep on the boat’s wooden seats. He eventually lived with a teacher from the base school, and then with the man who ran the lighthouse. Over the years, Mr. Romero worked as a janitor, firefighter and heavy-equipment operator. His favorite job was running and fixing the projector at the open-air base theater, the Lyceum. He retired with a small pension in 1998.

He built a dock among the mangroves behind his house and spends hours on a pontoon boat fishing for mackerel and downing cans of Miller Lite. He shies away from other exiles, wary of gossip. “I like to be by myself,” he said.

Capt. Nettleton worries what will become of the Cubans as they grow infirm. One exile was flown to the U.S. for dialysis. “I don’t envision us forcing them to go and leave the island, because of the commitment we previously made to them,” the captain said. “But it may come to that.”

Navy officials can’t find the documents that laid out the Kennedy administration’s original promise. The broad outlines have been passed down from one base commander to the next, reinforced by a 2006 law authorizing the Navy to “provide for the general welfare, including subsistence, housing, and health care” of the Cuban residents.

Life in Pre- and Post-Cold War Guantanamo Bay The gate separating the U.S. Naval base at Guantanamo Bay from the Cuban side, probably in the 1950s. U.S. Navy 1 of 10 • • • • • 1 of 10 Show Caption The gate separating the U.S. Naval base at Guantanamo Bay from the Cuban side, probably in the 1950s. U.S. Navy

Exile Phillip Gayle has married three times since moving to the base in 1965. He buried one wife and divorced another. He now shares a room with his third wife in the assisted-living facility the Navy built to care for the aging Cubans.

His wife, Felicita Gayle, is 95 years old. Her health troubles require round-the-clock aides. “She put light back in my life when my life was dim,” said Mr. Gayle, who is 80.

He works off his stress at Zumba classes in the base gym. An energetic Spanish-speaking instructor leads the exiles—one of them is in a wheelchair—in dance moves to the song, “I’m Sexy and I Know It.”

Mr. Gayle thinks often of the 3-year-old son he left behind when he fled Cuba. His son later joined the Cuban army and fought in Angola when Mr. Castro intervened to support the leftist government there.

“If Cuba is free and I die here, bury me over there,” Mr. Gayle said. “As long as Cuba isn’t free, bury me here.”

Phillip Gayle says the Castro regime pressed him to spy on the U.S. Instead he took refuge on the Naval base on Guantanamo Bay in 1965. Photo: Brandon Thibodeaux for The Wall Street Journal

The Naval cemetery is set amid underground ammunition bunkers. Among the hundreds of white-marble gravestones are 21 with the words “Cuban Exile” engraved beneath the names.

The U.S. continues to write the Cuban government an annual rent check of $4,085 for use of the base, a figure set in 1934. Each year, Havana signals its disapproval of the U.S. presence by refusing to cash it.

At one point in the 1960s, the Castro regime planted a wall of prickly plants to stop Cubans from seeking refuge on the base. The Americans dubbed it the Cactus Curtain.

In 1964, the Castro regime cut off water and power to the base. In response, President Lyndon B. Johnson dismissed most of the Cubans who commuted onto the facility. By that time, hundreds of Cubans—opponents of the Castro government—had already secured the Navy’s permission to live on the base indefinitely.

By 1987, the number of Cuban exiles on the base had dropped to 80. Today, there are 28 special category residents, including five Jamaicans who had won the status through marriage to exiled Cubans.

Capt. Nettleton estimates the Navy spends about $200,000 a year on salaries for civilian aides who assist the elderly Cubans. The Navy said it had no estimate of the cost of providing housing, utilities, medical care and other services for the exiles.

Cuban exiles gather for a prayer inside the Cuban Community Center on the base. Photo: Brandon Thibodeaux for The Wall Street Journal

Gloria Martinez, an 81-year-old cancer survivor, said she has stayed on the base in part because of the promise of lifetime care. Her late husband was a Cuban army sergeant in the 1950s who fought against insurgents led by Raúl Castro, Fidel Castro’s brother and now Cuba’s president.

In 1959, Ms. Martinez’s husband, Eduardo Martinez, got a job helping build the base bowling alley. Each night he returned to the Cuban side of the fence.

As the Castro regime cracked down on opponents, the couple’s house was repeatedly searched, Ms. Martinez recalled. She kept two hand grenades hidden in the house and said she had planned to use them if her husband ever faced a firing squad. She sealed her husband’s army medals in a jar and buried them under the patio.

“I told my husband they were looking for him, and he was never to come back to our home,” Ms. Martinez said.

One day in 1961, Mr. Martinez went to work on the base—and stayed. Ms. Martinez remained behind for a short time before making her own move. She glued her photo onto someone else’s ID card and headed to the base, repeating her fake name to herself: Jorgelina, Jorgelina. Jorgelina.

A portrait of the granddaughter of Gloria Martinez, an 81-year-old Cuban woman who has lived at the base since 1961. Mrs. Martinez’s granddaughter joined the U.S. Marine Corps. Photo: Brandon Thibodeaux for The Wall Street Journal

At a checkpoint, she recalled, the Cuban guard asked her name. Her mind went blank and she held up the ID card. “Ah, Jorgelina,” the guard said.

That was the last time she set foot on the Cuban side, she said. On the base she worked giving haircuts to Marines.

Their two children, now in their 40s and living in the U.S., were born on the base. Her husband died in 1988. When she fell ill with kidney cancer the following year, she was treated at a U.S. military hospital in Washington, D.C.

At the time, she worried if she moved to the U.S. she would lose her subsidized health care. Mostly, she said, she doesn’t want to leave Cuba, whose mountains are visible from her window. She has sisters living in Guantanamo City, an hour’s drive from her house, if such a drive were ever permitted.

“I’m close to my family here,” she said, “even though I don’t see them.”