With the Republican-controlled House and Democratic-controlled Senate at a stalemate, it was unclear how long the government would remain shuttered. The No. 2 Democrat in the Senate, Dick Durbin of Illinois, called the failure to pass a budget “conduct unbefitting a responsible Congress” and said he hoped it could be resolved by the end of the day Tuesday.

“Most people in the body politic are taking a look at this and saying, ‘A pox on both of your houses. It should never have reached this point,'” Durbin said Tuesday morning on CNN. “And there’s wisdom to that.”

The shutdown, the first since the winter of 1995-96, closed national parks, museums along the Washington Mall and the U.S. Capitol visitors center. The Smithsonian website displayed a red banner noting that “all Smithsonian museums and the National Zoo are closed.”

Agencies like NASA and the Environmental Protection Agency will be all but shuttered. People classified as essential government employees — such as air traffic controllers, Border Patrol agents and most food inspectors — will continue to work.

The health care law itself was unaffected as enrollment opened Tuesday for millions of people shopping for medical insurance.

The White House was operating with a skeletal staff, including household workers taking care of the first family’s residence and presidential aides working in the West Wing. A groundskeeper working outside Tuesday morning at daybreak said he was doing the job normally handled by four workers.

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 sought to delay key portions of the 2010 “Obamacare” law, then to delay 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?”

The underlying spending bill would fund the government through Nov. 15 if the Senate gets its way or until Dec. 15 if the House does.

Until now, such bills have been routinely passed with bipartisan support, ever since a pair of shutdowns 17 years ago engineered by then-Speaker Newt Gingrich severely damaged Republican election prospects and revived then-President Bill Clinton’s political standing.

Boehner had sought to avoid the shutdown and engineer passage of a “clean” temporary spending bill for averting a government shutdown.

This time tea party activists mobilized by freshman Sen. Ted Cruz, R-Texas, mounted a campaign to seize the must-do measure in an effort to derail Obamacare. GOP leaders voiced reservations and many Republican lawmakers predicted it wouldn’t work. Some even labeled it “stupid.”

But the success of Cruz and other tea party-endorsed conservatives who upset establishment GOP candidates in 2010 and 2012 primaries was a lesson learned for many Republican lawmakers going into next year’s election.

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Associated Press writer Lolita C. Baldor in Seoul, South Korea, contributed to this report.