The president and Mrs. Reagan began a marathon of ball-going last night with a visit to the American Legion's banquet in a capital celebrating both the inauguration and the hostages' freedom.

At the National Museum of Natural History there was sweet symbolism in the sight of the weathered old elephant who cast an ancient eye at the glittered multitudes surrounding him, but in the $2,000-apiece boxes that looked down upon the beast, confusion reigned among some toilers in the electoral vineyard.

The delegation from Poplar Bluff, Mo., sighed in weary unison when they discoverd that they had shelled out $3,200 for a nonexistent box, not to mention the gala tickets that had never arrived. "Well, it's been an experience, I'll say that much," said Mary McLane, who looked as wilted as the celery in her Bloody Mary. "All that money and it's standing room only." Still, said her colleague, Ron Douglas, "we have all the faith in the world in President Reagan; we'll forgive him this time."

Down below, the crowd was packed in close qurters to listen to Doc Severinsen, who sported a yellow ribbon on his rhinestoned blue tuxedo, and country singer Mel Tillis.

Looking on in quiet complacency was William Schneider, a state legislator from Wyoming. "People shouldn't look down on the inauguration for being expensive," he said. "I think the president should come across as a person who is given a little pomp and circumstance." On his finger he flashed a ring that bore the Romanoff eagle in diamonds set against a blue background. a"Fabrege, of course," he said. "I collect them. Where I come from, if the people can afford something, they ought to have it. Recession of course is not a new word to me, but it doesn't really affect Wyoming that much."

It was nearly midnight when President and Mrs. Reagan arrived. The crowd roared its approval as he spoke of his pelasure at seeing them there and gave them the most recent information about the hostages. As he left, smiles were evident. "It was worth the trip," one out-of-town guest said.

If there were those Washington denizens who left early, disappointed at the lack of famous faces, others were more easily satisfied. "Oh, look, honey," said one woman to her husband as she ascended on the escalator, "they rented that big old elephant just for us."