Congressional Democrats are marching in lockstep into the fourth week of a government shutdown, even as lawmakers brace for what could be the most painful point yet — a cutoff in federal food aid for 40 million people.

The U.S. government shut down this past Wednesday, furloughing any federal employees deemed non-essential and forcing the rest to work without pay (though they should receive backpay when the government is formally funded again). If the government reopens ...