WASHINGTON — Along Pennsylvania Avenue, not far from the Trump International Hotel and in between the Capitol and the White House, Carrie Wilder, a single mother and 11-year government employee, was on the verge of tears as she waited for free food.

“I’ve never felt the pervasive sense of hopelessness that I feel now,” said Ms. Wilder, who is doing clerical work without pay for the Justice Department during the government shutdown. She recently began asking her mother, 77, for money after she depleted savings she had accumulated since the government shutdown in 2013.

“I am watching the news every 10 minutes praying that something happens, because we cannot take any more,” said Ms. Wilder, one of many affected earning an annual salary of around $50,000.

She was among the federal workers who lined up on Wednesday at a kitchen opened by José Andrés, the Spanish Michelin-starred chef, disaster relief organizer and occasional antagonist of the Trump family. Operated by Mr. Andrés’s nonprofit organization, World Central Kitchen, the relief site is equipped to feed thousands of government workers and their families, many of whom are using food pantries for the first time as their savings accounts dwindle.