A shut down doesn't mean that all of the government will stop working. Any activities deemed not essential will cease. Things like national defense, etc will keep working.
Although it sounds big and scary, it's been more common than you may think. Since 1976, the US Government has been shut down 17 times.
The longest time was in December 1996. That shutdown lasted 21 days until Republicans and Bill Clinton agreed to a budget.
The shortest time has been 1 day and it has happened on four occasions. While most are because of a dispute, in October 1982, Congress simply didn't pass the required bills in time.
