The health care law itself was unaffected as enrollment opened Tuesday for millions of people shopping for medical insurance.
(Read more: 'Obamacare' exchanges start up)
The military will be paid under legislation freshly signed by Obama, but paychecks for other federal workers will be withheld until the impasse is broken. Federal workers were told to report to their jobs for a half-day but to perform only shutdown tasks like changing email greetings and closing down agencies' Internet sites.
The self-funded Postal Service will continue to operate and the government will continue to pay Social Security benefits and Medicare and Medicaid fees to doctors on time.
The Senate twice on Monday rejected House-passed bills that, first, conditioned keeping the government open to delaying key portions of the 2010 "Obamacare" law that take effect Tuesday, and then delayed for a year the law's requirement that millions of people buy medical insurance. The House passed the last version again early Tuesday; Senate Majority Leader Harry Reid said the same fate awaits it when the Senate reconvenes Tuesday morning.
"You don't get to extract a ransom for doing your job, for doing what you're supposed to be doing anyway, or just because there's a law there that you don't like," Obama said Monday, delivering a similar message in private phone calls later to Republican House Speaker John Boehner and other lawmakers.
Boehner said he didn't want a government shutdown, but added the health care law "is having a devastating impact. ... Something has to be done."
It wasn't clear how long the standoff would last, but it appeared that Obama and Reid had the upper hand.
"We can't win," said Sen. John McCain, R-Ariz., adding that "sooner or later" the House would have to agree to Democrats' demands for a simple, straightforward funding bill reopening the government.
The order directing federal agencies to "execute plans for an orderly shutdown due to the absence of appropriations" was issued by White House Budget Director Sylvia Burwell shortly before midnight Monday.
Around the same time, Obama appeared in a video message assuring members of the military they'll be paid under a law he just signed and telling civilian Defense Department employees that "you and your families deserve better than the dysfunction we're seeing in Congress."
Defense Secretary Chuck Hagel said Tuesday that Pentagon lawyers are trying to determine ways for some of the Defense Department's 400,000 furloughed civilians to continue working.
He bemoaned the standoff, telling reporters traveling with him in South Korea, "It does have an effect on our relationships around the world and it cuts straight to the obvious question: Can you rely on the United States as a reliable partner to fulfill its commitments to its allies?"