When the U.S. government shuts down, it doesn't really shut down entirely. Various vital services will continue to operate. Agencies like the Federal Aviation Administration and the Coast Guard continue to ensure our safety. The mail will still be delivered and food will still be inspected. However, many very important services will cease temporarily and federal facilities, such as the National Zoo, will be closed. For more details about which federal government functions will continue and which won't, see this page from the USA.gov site maintained by the General Services Administration:
Many government websites may also be unavailable during this time, but fortunately, USA.gov does not seem to be one of them.