TREASURE CAY, Little Abaco Island-For the chilled and weary vacation seeker who wants to get away from it all without traveling too far from Washington, there's no denying the attraction of the Bahamas' "Family Islands."

Of the 700 or so smaller islands and cays comprising the Bahamian archipelago, only a handful are now prepared to accommodate tourists -- among them Andros, the Exumas, Eleuthera, San Salvador, Bimini and the Abacos.

But the welcome is warm, the mood is low-keyed and restful, and the solitary beaches can run on for miles.

One reason for the peace and quiet is the small number of guests, about 15 percent of the Bahamas' total, which was slightly under 2 million visitors in 1980, a very good tourism year here.

"It's a matter of logistics," explained Colin Tatem of the Princess Hotels. "The average distance between islands is between 50 and 80 miles, and that makes transportation expensive."

Several airlines and air charter services fly to a dozen or so of the Family Islands. Bahamasair offers one-day excursions to several, but it is necessary to return to Nassau each time before taking off again for a new island destination. Mailboats, ferries and chartered planes are other ways to make an out-island trip. Many visitors arrive on cruise ships that provide accommodations and meals while spending from a few hours to more than a day anchored at these isolated flat, rocky coral formations.

Large Miami-based vessels like the Norwegian Caribbean Line's Norway, Southward, Starward and Sunward II spend the better part of a day anchored offshore at places like Great Stirrup Cay and Little San Salvador (in January NCL's Skyward will be included), bringing passengers ashore in launches for sunbathing on the beach, shelling, swimming and a barbecue. The American Canadian Line's smaller New Shoreham II, which can pull right up on a beach and lower its gangplank, has just begun its sixth season with 12-day cruises from Nassau and Freeport to Abaco, Eleuthera, Berry Islands, Crown Haven, Sampson's Cay, Staniel Cay, Halls Pond Cay and Highbourne Cay (not all included in one cruise).

Granted the somewhat higher initial cost just to get there, what makes the Family Islands so attractive? For me it's their beauty and tranquility, the opportunity to enjoy companionship or savor solitude.

A 30-minute trip with Bahamasair from Nassau brought me to South Andros. This flight makes the run five times a week. Neville Lee Choy, manager of the Las Palmas Beach Hotel where I was to stay (the only hotel on that part of the island), has already become a local legend. As soon as Choy hears the engines of an incoming plane overhead, he jumps into his car to fetch his guests from the airport.

Operated by Resorts International, Las Palmas is a one-story, 20-room hotel set on five miles of white sand beach. Each room fronts on the beach. Prices range from $40 for a single to $50 double, winter season rates. Add a reasonable $11 per person for a very satisfying breakfast and dinner selected from a wide choice of Bahamian, American and Continental cuisines. These prices are typical for a small, moderately priced Family Islands hotel.

Telephones? There are two "communications centers" two and eight miles from the hotel. Television? A set sits -- for the most part ignored -- in the hotel lobby. Electric power? The island has none. Two generators in the hotel provide for all its needs. Medical assistance? A physician from a nearby clinic is on 24-hour call.

"You need no hotel room keys here," Choy told me, emphasizing the sense of security. In fact, the only cautionary sign I saw anywhere on my out-islands travels stood near some trees on the South Andros beach. It bore the message: "WATCH OUT FOR FALLING COCONUTS!"

For all their distance from the relatively fast pace of Nassau and Freeport, the Family Islands offer sufficient activity to fill the hours: swimming, snorkeling, scuba diving, sunbathing, fishing, sailing, boat trips to nearby cays and reefs, biking and hiking along miles of open beach. Each island has one or more special attractions of its own. Andros, for example, offers boat trips, with scuba diving available, to what is touted as the world's second-biggest reef after Australia's Great Barrier Reef.

In 1980, the Bahamas Ministry of Tourism started a vigorous promotion campaign to introduce foreign travelers as well as Bahamians to the Family Islands. Vincent Vanderpool-Wallace of the Ministry of Tourism in Nassau assured me: "The Family Islands have abundant accommodations capacity. At this point they are a kind of stepchildren. They appeal especially to an older group, to the more sophisticated traveler."

I returned to Nassau and then flew to Treasure Cay, a one-hour trip. Hugging the eastern shore of Little Abaco Island, Treasure Cay is precisely what Vanderpool-Wallace had in mind. Leonard Thompson, who now operates the Great Abaco Hotel at nearby Marsh Harbour, developed the multimillion-dollar Treasure Cay resort complex out of a vegetable and fruit farm 18 years ago. "When we started Treasure Cay," Thompson recalled, "there was no road between it and Marsh Harbour. We had to go by boat. We put in an airstrip and a terminal building."

Today the resort includes the two-story Treasure Cay Beach Hotel and 56 rental villas accommodating about 400 guests. Winter rates range from $72 single to $99 double. The villas are $99 single, $111 double. Dinner entre'e prices range from $12.50 for Bahamian Cracked Conch to $14.50 for Deep Fried Fantail Shrimp. Dessert is $2, beverage $1, plus the usual 15-percent gratuity. The hotel does have its own telephone.

Treasure Cay offers everything in water sports, including a branch of Freeport's UNEXSO for scuba-diving lessons and reef dives. Golfing, tennis and biking are standard diversions, while the resort's "special" is a boat trip to Green Turtle Cay and its 18th-century, New England-style village of New Plymouth, a legacy of British Loyalists who migrated south under pressure of events in their earlier mainland home far to the north.

"Treasure Cay is for the upper class largely, those of quiet wealth," said Berkley Evans, president of Treasure Cay Ltd., confirming my own observations. "Most of our visitors are middle-aged. They come with their children and grandchildren."

Everett Pinder, general manager of Treasure Cay Beach Hotel and Villas, elaborated on the importance of the area's peaceful, quiet atmosphere. "It's not the regimented type of place, where guests get an activity schedule the minute they sign in. You do your own thing here. They just get a map and information on all the sports opportunities."

Nevertheless, there's change in the air, even at Treasure Cay. Recently a $3-million airport was completed here, and already it is feared the new terminal will be too small by the end of 1982. Plans are under way to include Treasure Cay on the Bahamasair run from Atlanta to Nassau, and for Air Florida to Sandwich in Treasure Cay on its flight from New York to Miami.

New construction is also in sight. The present 45-boat marina will be expanded to accommodate 70 boats in 1982. The tract next to Treasure Cay Beach Hotel is being cleared to make way for a 300-room hotel, together with restaurants and shops, now scheduled for completion in late 1983 or early 1984. There's talk, too, of adding 70 to 80 rooms to the present hotel. Condos, town houses, and time-sharing units have also sprung up. More are planned for the future.

"But we cannot destroy the ambience of Treasure Cay," declared Pinder, well aware of the impact runaway development could have on the resort's prized peace and quiet. "Part of our planning is to keep that foremost in our thinking. We're committed to that. You're left alone, you walk a few steps down the beach, and you feel you're the only person for miles around. We want to keep it that way."

South Andros is expecting change next year, too. Electricity and phone service will be available. "Many people here don't like it," Choy told me, "because their income is from the fishing industry alone. Running your house by a generator is a lot more economical; you turn it on only when you need it."

Change always has a price. I do hope it's kind to the Family Islands. And I hope that if I visit the Family Islands again some day, I can still walk along a beach for miles without meeting anyone else, and still will be able to tell the following story.

When Simpson, my Treasure Cay guide, was driving me around the resort's single road, he assured me I needn't worry about reconfirmation of my flight home. By coincidence, he saw the Air Florida agent walking along the road. "Hey, Maxine," he called out to her in mock concern, "this lady needs reconfirmation for returning to Miami tomorrow."

"There's always space on the return flights," she called back with a reassuring smile and continued on her way.

For further information on the Family Islands or any other part of the Bahamas, contact the Bahamas Tourist Office, 1730 Rhode Island Ave., NW, Washington 20036 (202-659-9135). For airline, cruise ship or hotel reservations, see your travel agent.