Luckily,
Currently they are:
The ongoing satellite missions are also exempt, but work on planned missions will stop.
Here's the wording from Elizabeth Robinson NASA's Chief Financial Officer:
There are two major operations or classes of operations that would require ongoing support in accordance with the definitions of excepted activities identified above. First NASA currently is operating the ISS with a crew of 6 astronauts/cosmonauts, which has been in continuous operation since 1998. To protect the life of the crew as well as the assets themselves, we would continue to support planned operations of the iSS during any funding hiatus. Moreover, NASA will be closely monitoring the impact of an extended shutdown to determine if crew transportation or cargo resupply services are required to mitigate imminent threats to life and property on the ISS or other areas.
Second, if a satellite mission is in the operations phase, we will maintain operations that are essential to ensure the safety of that satellite and the data received from it. However if a satellite mission has not yet been launched, work will generally cease on that project.
NASA has 18,000 employees, and only about 600 would be able to continue working according to Space.com's Mike Wall. Things like NASA TV and website, NASA educational support, and tours of NASA centers, will be stopped during the shut down.
Her letter also says that the shut down process should take about half a day.