On the second day of the year, Danielle Miller gave up on the federal government.

Furloughed from her Internal Revenue Service job near Cincinnati and fearful of running out of money during the partial government shutdown, she filed for unemployment benefits: $414 a week, about $200 less than usual.

“Once Christmas came and went, after New Year’s, I was like, I can’t go on,” said Ms. Miller, a single mother who has worked for the I.R.S. for almost 14 years. She spent part of this week calculating when her first unemployment check would arrive. “It’s disappointing, and it’s frustrating,” she said. “I have a job.”

The shutdown, the longest on record, is prompting tens of thousands of federal employees to seek jobless benefits. As the impasse meanders through its fourth week and more bills come due, their numbers have been growing.

On Thursday, two days after the White House doubled its projections and warned that the shutdown was reducing quarterly economic growth by 0.13 percentage points per week, the Labor Department reported 10,454 initial claims by federal workers for the week that ended Jan. 5, doubling the previous week’s figure. Thousands more have applied since, state officials said.