District of Columbia Government Shutdown Avoidance Act of 2013 - Amends the District of Columbia Home Rule Act to authorize the District of Columbia to obligate and expend local funds for projects and activities in accordance with the local budget act for the fiscal year, at the rate of operations provided under that act, during any portion of a fiscal year in which neither the regular District of Columbia appropriation bill for such fiscal year nor a District of Columbia continuing resolution for it is in effect.