Storage and Transportation Of Residual and Excess Nuclear Fuel Act of 2019 or the STORE Nuclear Fuel Act of 2019 This bill directs the Department of Energy (DOE) to establish a program under which it may (1) site, construct, and operate one or more consolidated interim storage facilities for the storage of spent nuclear fuel and high-level radioactive waste; and (2) enter into contracts with the licensee of a private interim storage facility in order to take title to, transport, and store in it either high-level radioactive waste or spent nuclear fuel. DOE must prioritize contracting with private facilities unless it can demonstrate that it is able to develop a site in a more cost-effective manner.