Visa Overstay Enforcement Act of 2013 - Amends the Immigration and Nationality Act to make it a felony for any alien to be unlawfully present in the United States for a period of 30 consecutive days, except because of illness or any other extenuating circumstance. Requires such an alien to be punished: (1) in the case of a first offense, with a fine of not more than $10,000 or by imprisonment for not more than one year, or both; and (2) in the case of a subsequent offense, with a fine of not more than $15,000 or by imprisonment for not more than five years, or both.

Declares that an alien convicted: (1) of a first offense may not be admitted to the United States for 5 years, and may not be granted a visa for 10 years; and (2) of a subsequent offense may not be admitted to the United States, and may not be granted a visa.

Directs the the Secretary of Homeland Security (DHS) to: (1) submit to Congress within 90 days a plan to implement a biometric exit capability using a person's fingerprints at all land, sea, and air ports of entry under the US-VISIT program; and (2) establish the biometric entry and exit data system required by the Intelligence Reform and Terrorism Prevention Act of 2004 within one year at each airport that is a U.S. port of entry, and within two years at all U.S. ports of entry.