Family Health Care Accessibility Act of 2013 - Amends the Public Health Service Act to deem a health professional volunteer providing primary health care to an individual at a community health center to be an employee of the Public Health Service for purposes of any civil action that may arise from providing services to patients. Sets forth conditions for such liability protection, including: (1) the service is provided to the individual at a community health center or through offsite programs or events carried out by such center; and (2) the health care practitioner does not receive any compensation for providing the service, except repayment for reasonable expenses.

Considers an entity as sponsoring the health care practitioner if the entity submits an application to the Secretary of Health and Human Services (HHS), and the Secretary determines that the health care practitioner is deemed to be an employee of the Public Health Service.

Requires the Attorney General to submit to Congress an estimate of the amount of claims (together with related fees and expenses of witnesses) that, by reason of the actions or omissions of health professional volunteers, will be paid pursuant to this Act annually. Requires the Secretary to transfer such estimated amount from the claims fund to the appropriate accounts in the Treasury, subject to the extent of amounts in the fund.

Makes this Act effective on October 1, 2014.