Fulfilling Our Responsibility for Efficient and Sustainable Timber Supply Act of 2013 or the FORESTS Act of 2013 - Extends through FY2023 the authority for the Forest Service and the Bureau of Land Management (BLM) to enter into stewardship end result contracting projects with private persons or other public or private entities to perform services to achieve land management goals for national forests and public lands that meet local and rural community needs.

Prohibits a multiyear stewardship contract from exceeding a 20-year period (under current law, a 10-year period).

Bars the Secretary of Agriculture (USDA) and the Secretary of the Interior (the Secretaries) from being required to obligate funds to cover the costs of cancelling or terminating multiyear stewardship contracts or agreements until they are cancelled or terminated. Specifies the funding sources from which the costs of cancelling or terminating a multiyear stewardship contract may be paid.

Requires 25% of timber sale receipts from a multiyear stewardship contract or agreement to be paid to the county within whose boundaries the receipts are derived.

Authorizes the Secretaries to enter into cooperative agreements and contracts with state foresters to provide forest, rangeland, and watershed restoration and protection services. Permits state foresters to enter into subcontracts to provide such restoration and protection services. Prohibits the extension of such an agreement or contract beyond a 20-year period.