As you may already be aware, the U.S. government isn't exactly firing on all cylinders right now. NASA's website is offline, as are many websites that end on the .gov domain. But if all that has you worried, rest assured: all is not lost.
For starters, even though it looks like NASA has closed the door and thrown away the key (for now), the agency has actually been able to keep a couple folks at their desks. Specifically, they're the folks keeping tabs on the International Space Station (ISS) and its resident astronauts. You can rest assured that the ISS isn't going to go all Europa Report while the government is down.
As for the Curiosity rover, it's true that it won't be going anywhere for now. Curiosity has been put into sleep mode. While waking it up again will be a little more complicated than just tapping a space bar, the rover is actually very safe this way. Incidentally, NASA figured out a way for Hubble and the rest of our currently operating spacecraft to stay up and running, too, so we can still get our fix of awesome nebulae and supernovas.
There's good news over at the National Science Foundation too. A lot of the scientists who have already received funding through the foundation aren't federal employees, so they've got no need to stop their research. New funding is on hold, but as with the rest of the suspended services, that's just until things get back up and running.
And that's the real story here: things are tight right now, but where it can and where it needs to be, the government is still running. We've got a hunch this whole shutdown thing isn't going to last that long. Think of it as a national tornado advisory: pop in a couple of old movies, stay safe and it'll be over before you know it.