January 3, 2014
If you missed the initial enrollment phase (October 1, 2013 - December 23, 2013), no worries. You can still sign up for health insurance until March 31, 2014. After that, open enrollment will close until October, 2014 when enrollment will open for 2015.
If you sign up now through January 15, 2014, your insurance will begin February 1, 2014. For more information on the time frame for enrolling and insurance coverage beginning, please visit, https://www.healthcare.gov/what-key-dates-do-i-need-to-know/#part=3