Well, as this is a law blog, I'll focus on the judicial branch. Our beloved federal courts (aka Article III courts) are part of our federal government. So, do they shut down if the federal government shuts down? The Blog of Legal Times reports that the Judiciary Could Limp Through Shutdown - for a While:
[T]he federal judiciary says there should be no visible disruption in its operations for two weeks. The judiciary pays its bills in part with fees, which are outside the regular appropriations process, and it says it has enough in reserves to keep its doors open even if Congress does not agree on a budget.An extended shutdown could require individual courts, and even individual judges, to make determinations about which functions are "essential." But, in the short run, it's "on with the show."
Posted by Philip Miles, an attorney with McQuaide Blasko in State College, Pennsylvania in the firm's civil litigation and labor and employment law practice groups.