The diplomatic outrage triggered by the Soviet downing of a South Korean airliner was transformed into quiet anger yesterday inside the cool stone solemnity of Washington Cathedral.

In a 26-minute memorial service in the cavernous Gothic church, President Reagan, along with a legion of foreign diplomats, members of Congress and several Cabinet members, gathered with about 60 family members of victims to mourn the 269 who were killed Sept. 1 when a Soviet missile destroyed Korean Air Lines Flight 007.

With dark-jacketed diplomats in the front of the cathedral's main nave and pastel-clad tourists in the rear, a congregation of 750 intoned the "Yea, though I walk through the valley of the shadow of death" response from the 23rd Psalm, sang Martin Luther's "A Mighty Fortress," and listened soberly as the Episcopal Bishop of Washington rebuked the Soviet Union.

"We would like to find some means of justification, some possibility of getting the Soviet Union 'off the hook' so to speak. We have not been able to do so nor have they offered any help," said the Rt. Rev. John T. Walker. "There is no way to justify the destruction of helpless, innocent people."

Reagan, praised in the bishop's homily for his "controlled anger" at the Soviets, sat with his wife, Nancy, in the front pew of the church, across the aisle from the families of victims.

After the service, the president met with them in an adjoining chapel. He shook hands with each family member, including Tryggvi McDonald, the son of Rep. Larry McDonald (D-Ga.), a passenger on the downed jet. According to deputy press secretary Larry Speakes, Reagan told them:

"No words can compensate for the burden of sorrow you carry. At times like this we can only trust in God for his mercy and wisdom. We are determined to do everything we can to see if things can't be done so that events like this never happen again. I promise we'll do everything we can."

Back in the White House, Reagan proclaimed Sunday a national day of mourning. The proclamation said, in part, "September 1, 1983, will be seared in the minds of civilized people everywhere as the night of the Korean Air Lines Massacre. . . . This is a crime against humanity that must never be forgotten, here or throughout the world."

Reagan said his proclamation was the sixth national day of mourning in history. However, Sunday in fact will mark the seventh national day of mourning in the past 20 years. Other days have been set aside for the deaths of presidents Kennedy, Eisenhower, Truman and Johnson, as well as Martin Luther King Jr. and Robert F. Kennedy.

Among the estimated 150 foreign diplomats at the memorial service were the ambassadors of five nations--Korea, Japan, Canada, Sweden and Thailand--whose citizens died aboard the South Korean airliner. Also in attendance were lower-ranking diplomats from Australia, Malaysia, India and the Philippines, countries that also lost citizens in the disaster.

The State Department, which came up with the idea for a memorial service, gave its employes two hours off yesterday to attended the noontime service.

In addition to the dignitaries and government officials, there were several hundred people who attended the service out of their own sense of obligation to mourn those who died.

"I came just to be a part of consoling the families," said George Berry, of Lafayette, La., who took time out from his vacation to attend the service.