Great. The economy teeters somewhere between a fender-bender caused by 800,000 people suddenly not collecting paychecks and the likely fatal highway pile-up that would ensue if Washington defaults on Oct. 17 — and “responsible” Republicans are wandering around Capitol Hill looking like dazed and confused accident victims.

Republican leaders had vainly hoped that the rebel-right faction demanding a defunding of Obamacare in return for funding the government would peter out. That Ted Cruz in the Senate and Tom Graves in the House would fold up their cards and go home. Ain’t gonna happen.

Now that same wishful thinking is being applied to crafting possible exit strategies. Republican leaders hope that if President Obama won’t cave on changing his signature legislation, the Affordable Care Act, he might agree to hold down costs on Social Security and Medicare instead. Then enough Republicans would come on board to pass a budget bill, open the government, and raise the debt ceiling. Twin crises averted.

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Would Obama agree to this? In an alternative political universe, maybe. The President supports means-testing for Medicare and is willing to at least consider changing the formula for raises in Social Security checks. But for all the talk of “grand bargains” in the past, they have fizzled over Obama’s reluctance to stand down fierce opposition from within his liberal base and from powerful interest groups like the AARP. Now, with Republicans being blamed for the government shutdown, the President has even less incentive to negotiate.

And here’s the irony: Even if the White House agreed to a deal, the GOP rightists who started this fight over Obamacare probably won’t support an exit strategy that instead only relies on entitlement concessions. “The guys that got us into this mess won’t even be there when we cut a deal to get us out,” says the same member of Congress.

In fact, the right’s activists want nothing short of defunding Obamacare. As RedState’s Erick Erickson wrote this week: “Republicans should hold the line. They should demand that ObamaCare be defunded. They are going to get the blame for a shutdown. They might as well own it and keep it shut down until they get their way. Sure, it will take time, but eventually Barack Obama will blink.’’

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So the nation’s economy is being held hostage to the dubious proposition that Barack Obama, comfortably reelected 11 months ago, will “blink” over Obamacare. No wonder nerves are fraying on both sides of the aisle. One Democratic member said he worries that this clash is far more serious than any of the cliff-hanging budget and debt battles over the past three years. “It feels sketchier than before,” he says. “The stakes keep getting higher.” Keep in mind, too, that during the extended government shutdowns in the 1990s, the economy was much stronger.

Just weeks ago, vocal Republican leaders in both Houses were decrying the right’s defund-Obamacare campaign as “silly” and “dumb” — a self-destructive fight that couldn’t be won with Democrats controlling both the Senate and the White House. But House Speaker John Boehner acceded to his right flank’s demands — and now finds his party mired in that very same unwinnable fight.

In Fortune’s Sept. 16 issue, I likened the GOP’s refuseniks to the obstinate little boy Pierre, from the Maurice Sendak children’s tale. Pierre shouts “I don’t care” at every reasonable request made of him — and repeats it again when a lion threatens to eat him.

Now the lion has swallowed not only the Pierre wing of the party but Republican congressional leaders as well. In Sendak’s tale, Pierre’s parents rescue him in the end. It’s not clear who will come to the rescue this time.