All 6,250 shareholders of Cook Inlet Region Inc. are Alaskan Indians, but forget dog sleds and raw fish: They're very rich Indians.

"These are not a bunch of people running around in furs," said Thomas A. Morehouse of the University of Alaska's Institute of Social and Economic Research.

Cook Inlet is, instead, a financial powerhouse, the richest of 13 regional native corporations established by Congress in 1971 to repay Indians and Eskimos for lands taken by the federal government.

Today, the group boasts a vast array of holdings, including lucrative oil, gas and mineral rights, drilling ventures, a construction company, real estate, a television station -- and two prime properties in Northern Virginia.

Using a system of credits granted under the 1971 act, Cook Inlet two years ago acquired the old Ford Motor Co. plant, a key part of Alexandria's developing waterfront, for $14.2 million.

Last week, the group surfaced here again, bidding $10.1 million to take over a 107-acre parcel at Fort Belvoir in southeastern Fairfax County.

There are plans to develop both sites as commercial ventures.

"If you want to know what they're interested in," said one Interior official in Anchorage last week, "the answer is, 'Anything they think they can make money on.' "

Although few details are available about their plans for the two Virginia sites, their dealings here so far have been amicable.

"I think they're going to be quite cooperative," said Alexandria Mayor James Moran. "I've talked with them. I like them."

"They were advised to talk with local officials and groups," said retired Rear Adm. Kleber Masterson Jr., president of the Old Town Civic Association, "and they did."

As a result, Cook Inlet appears to have a three-part image here: friendly, highly professional and a little mysterious.

They are a shining success story in the Alaska economy. While some of the native corporations suffered high-risk failures and others have fared moderately well, Cook Inlet, operating with a lean staff of 55, has prospered.

According to an annual report on file with the state of Alaska, the group had revenue in fiscal 1984 of $27 million and profits of $18.6 million.

Soon to be released figures for fiscal 1985 will show profits of about $26 million. "It was another good year," said Roy Huhndorf, the native Alaskan who serves as Cook Inlet's president.

With its federal credits, Cook Inlet has bid successfully at General Services Administration auctions for dozens of surplus properties across the nation. It owns a former Nike missile site, an old NASA building in New Orleans, a former GSA motor pool in Tampa, Fla., a piece of Lackland Air Force Base in Texas, a former Army Corps of Engineers property in Miami Beach and other land in Alaska, southern California, Denver, Minneapolis and Honolulu.

In addition, there are other private joint ventures "in every phase of Alaska's economy," said Robert Arndorfer, an Interior Department official in Anchorage.

Last year, the corporation headed a group that bought WTNH-TV, the ABC affiliate serving New Haven and Hartford, Conn., for about $170 million.

The Anchorage Daily News has valued Cook Inlet's assets at about $200 million, not including oil and gas interests. And because of losses in earlier years, mainly through resource depletion allowances, the corporation has yet to pay a dime in taxes.

The wealth has its roots in the 1971 Alaska Native Claims Settlement Act under which, ironically, Cook Inlet started out at a disadvantage.

Designed to allow native groups to reclaim their aboriginal lands, the law offered little at first to the Cook Inlet Indians. Their ancestral lands already were highly developed, with a sizable portion paved over to create the city of Anchorage.

Offered other land described by one official as "mountaintops and glaciers," the natives sued the Interior Department. In a settlement in the mid-1970s, Interior granted the Cook Inlet natives $78 million in cash and entitlements to 2.3 million acres of Alaskan tundra, valued initially at $500 an acre.

Cook Inlet Region Inc. became the only native corporation allowed to bid for federal surplus property outside Alaska, converting some of the undeveloped wilderness to credits.

Today, with adjustments for inflation, the credits total about $110 million. About $55 million has been used for acquisitions like the Alexandria and Fort Belvoir sites, according to federal officials.

Some other corporations that did acquire much or all of their native lands have fared less well.

One, Bering Straits Native Corp., filed last week for reorganization under bankruptcy laws. "Bering Straits is a cold, lonely place," said an Interior official, "and all they got was rock, rock and more rock."

Each Cook Inlet native originally received 100 shares in the corporation. Through deaths and inheritance, some natives now hold 200 shares.

Earnings from Cook Inlet's business ventures have swelled the equity held by its native shareholders to $128 million -- about $205 a share -- according to the 1984 annual report.

Congress now is considering whether native shareholders will be allowed to cash in their interests when a 20-year prohibition on such sales expires in 1991. That possibility has raised fears of takeover attempts.

Cook Inlet "keeps much of its oil and gas information private to be in an advantageous position to fend off a T. Boone Pickens in 1992," said Ron Chappell, an Anchorage Daily News reporter who covers the corporations.

The management draws praise from state officials and interested scholars, who describe the group's executives as savvy and sophisticated.

Cook Inlet President Huhndorf, who grew up in a Yukon River village and whose mother was a Cook Inlet Indian, earned a business degree from the University of Alaska while overseeing the corporation. He serves now as a regent of the university.

"We're selective about which properties we bid on," said Huhndorf. "We look at the local economy and at what cycle the local real-estate industry is in. We look for growth potential, the same as any other investor."

Their secret, according to a U.S. Bureau of Land Management official in Anchorage, is twofold: "One, they have real good management, and two, God was real good to them in the first place. Theirs happened to be the place where the whites settled. That was strictly fortuitous." CAPTION: Picture, Old Ford Motor Co. plant along Alexandria waterfront that Cook Inlet Indians bought two years ago for $14.2 million.