The federal government is currently engaged in an out-of-control spiral of debt. They keep just raising the debt ceiling so they can take out more debt. It's the same pattern of behavior of a drug addict or alcoholic constantly needing a stronger dose. Eventually the debt will become unsustainable and the bottom will fall out of the whole system. The government will have no choice but to shut down for the simple fact that there will be no money to fund it.
A complete government shutdown could happen. It would not be functional and the country would be in complete disarray but it could happen. Without government social problems, court systems, military, and other forms of government, society would be forced to come up with a new system of doing things. It could be done but it would be a complete mess.
Without order, people will revert to their instincts. There will be no police, all money becomes worthless, and people will just be rushing into supermarkets and taking whatever. People will also have less motivation to work, and that means no doctors, no lawyers (or law), and no factories, electricity, or any of the other things we take for granted. People will steal medicine, overdose, kill people, etc. Humankind as we know it today will have been eliminated within hours.