The Smithsonian, which employs more than 4,000 people, would have to cut that number down to just 688. The Smithsonian Institute, which includes some of the country's most popular museums, will not be able to keep its doors open to the public.
Workers who will be furloughed will be "hurt greatly," Obama said Monday in a statement at the White House.
"And as a consequence all of us will be hurt greatly, should Congress choose to shut the people's government down," he added.
Obama also warned lawmakers that a shutdown would "throw a wrench into the gears of our economy."
"The idea of putting the American people's hard-earned progress at risk is the height of irresponsibility, and it doesn't have to happen," the president said. "Let me repeat this. It does not have to happen."
U.S. troops were shielded from any damage to their wallets when Obama signed legislation assuring the military would be paid in the in the event of a shutdown.
ABC News' John Parkinson and Mary Bruce contributed to this report.