Hagel, who oversees as much as half of the government civilians who would be furloughed next week if Congress doesn't reach a budget agreement, said the impasse threatens to delay paychecks to troops serving in Afghanistan.
"When you look at the greatest democracy in the world, the largest economy in the world and we're putting our people through this -- that's not leadership. That's abdication of responsibilities," Hagel said on his plane en route to South Korea. "This is an astoundingly irresponsible way to govern."
Hagel added: "It is really dangerously shortsighted and irresponsible, because what this will lead to in the United States of America, if this continues, is we will have a country that's ungovernable."
Roughly 400,000 Defense Department civilians could face furloughs if Congress fails to agree on a short-term spending plan to keep the government running. And while military troops would continue to work, and eventually be paid, a shutdown would delay their pay until funding was restored.
There are about 800,000 civilians in the department and 1.4 million active-duty military members. About half of the civilians would be exempt from furloughs, and they would only be paid retroactively if the legislation specifies it.
Senators on Friday sent legislation back to the House that would keep the government's doors open until Nov. 15. Majority Democrats removed a provision to defund the Affordable Care Act, also called Obamacare. But House Republicans are fractured, with tea partyers insisting that they will not support a measure than includes funding for President Barack Obama's health care law.
Hagel, a former senator who served as an enlisted soldier in the Vietnam War, said lawmakers need to stop and think about the young service members who may be married and have one or two children at home. A lot of them, he said, don't have a lot of excess money and going without a paycheck, even for a short time, would be difficult.
Hagel said much of his time in recent weeks has been spent in meetings with his budget staff as they struggle with plans for the government shutdown while also trying to identify more savings in order to meet looming budget cuts.
Pentagon Comptroller Robert Hale, in a Friday briefing, said the Pentagon will be able to launch any military operations ordered by the president. Asked whether a military strike into Syria, for example, could be conducted under the shutdown, Hale said yes, if ordered by senior leaders.
Combat units, including troops in Afghanistan or ships in the Mediterranean Sea, can continue to conduct activities and training that are part of any military operation. Other training not tied to a specific operation would not go on.
Hale said it's not yet clear whether the shutdown would impede the ability of service members leaving Afghanistan as part of the troop drawdown. But those decisions would be made by commanders on the ground.
Hagel, who is heading to meetings with top defense and diplomatic leaders in South Korea and Japan, said it is important that they know that the impasse won't affect U.S. relations with other countries.
"I think it's very, very important that we continue to assure our allies in this region of the world that we are committed to these alliances and, yes, we have the ups and downs like democracies do and we'll work our way through these domestic challenges," he said.
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