The Alfalfa Club, that indescribable very private stag organization, that meets annually for the sole purpose of eating, drinking and telling political jokes, Saturday night nominated Sen. Sam Nunn (D-Ga.) for president of the United States. He will run on the Alfalfa ticket.

His 500 or so nominators were an impressive crew, an array of faces often seen on the front pages -- Vice President George Bush as well as most of the Cabinet; the chairmen of some of the largest corporations in the country as well as many former household names.

So far only one Alfalfa nominee has ever made it to the White House -- Ronald Reagan. Nunn, however, was quite optimistic, according to sources, though the same sources were quick to note that Nunn's one-liners were of such high professional quality that perhaps the senator should consider a new career. "That guy should head for Vegas to do a show," said one ("don't use my name") dinner guest.

As usual, the 69th annual black-tie affair was OFF THE RECORD. Even the sight of a reporter and photographer at the Capital Hilton made guests, some wearing Alfalfa medallions and others wearing patent-leather dinner shoes with little bows, visibly nervous.

However, several eyewitnesses offered to relate some of the remarks if quoted anonymously.

In his acceptance speech, Nunn:

* Said he planned on throwing a dinner for minorities and inviting Mike Wallace to be the caterer;

* Commended Treasury Secretary Donald Regan on being a "conscientious objector in the war on poverty";

* Asserted that former president Carter did to America what Sherman did to Atlanta;

* Praised Richard V. Allen for being the first American to beat the Japanese on a business deal.

The evening started out with cocktails, black limos, Secret Service and a big crowd in the hotel lobby. The biggest applause went to George Bush, Sen. Barry Goldwater (R-Ariz.), Douglas Fairbanks Jr. and Sen. Ted Kennedy (D-Mass.).

"Oh, he's just so-o-o handsome," sighed one woman of Kennedy. She described herself as a "little ol' gal from Oklahoma."

"He sure is," drawled her friend. "But look -- his hair is getting gray."

Secretary of State Alexander Haig arrived with a motorcade, and nearly as many Secret Service agents as Bush. Presidential aides Edwin Meese and Mike Deaver also were accompanied by agents.

Senate Majority Leader Howard Baker, a Republican, came with former HEW secretary Joe Califano, a Democrat. As incoming president of the club, Baker was asked to share his remarks with the press. "I just caaaan't," he drawled with firm resolve.

Outgoing president Henry Kissinger also refused to share his remarks, but, sources say, he was almost, but not quite, as funny as Nunn. "I would have preferred to read my memoirs," Kissinger was quoted as saying, "but the Hilton has this room booked for next Tuesday. I could shorten my book by one-third . . . by removing all the I's . . ."

Former defense secretary Donald Rumsfeld flew in from Chicago for the affair "to see some old friends."

"My wife goes to affairs where only women go," said former secretary of state Ed Muskie, "and I don't mind, so why not? Actually, she gives me hell."

The Alfalfa Club, started in 1913, always has 150 members. The only way someone new can join, says a member, is if "someone dies."

Last night, it was unclear who had died, but Robert Strauss, Treasury Secretary Regan, and House Minority Leader Robert H. Michel (R-Ill.) were invited to join the club.

Other members include former attorney general Griffin Bell; Sen. Lloyd Bentsen (D-Tex.); Chief Justice Warren Burger; Jay Rockefeller, governor of West Virginia; attorney Edward Bennett Williams; and FBI director William Webster.

Each member was allowed to bring two guests to the dinner, which consisted of lobster appetizer, filet mignon and baked Alaska. The wine flowed heavy. The Marine Band played on.