1. A storage system, comprising: a first virtual volume that is constituted by a plurality of virtual areas and is based on thin provisioning technology; a second virtual volume that is based on storage virtualization technology and corresponds with an external logical volume that is formed based on a removable external storage device; a pool that has a plurality of pages which are storage areas that can be assigned to the virtual areas of the first virtual volume; mapping information that indicates which of the plurality of pages correspond with which virtual areas; an input/output (I/O) processing unit that receives a write request from a computer, assigns a page in the pool to a virtual area of the first virtual volume that is designated by the write request, writes data fragments corresponding to the write request to the assigned page, and associates information representing the page with information representing the virtual area to which the page is assigned, in the mapping information; and a backup processing unit that copies a plurality of data fragments that are stored in each of a plurality of pages that are assigned to the first virtual volume to the second virtual volume, the plurality of data fragments being serially arranged in the second virtual volume, updates information representing the respective copy source pages in the mapping information to information that represents respective copy destination pages for the data fragments stored in the respective copy source pages, and copies the updated mapping information to the second virtual volume, wherein the plurality of data fragments are serially arranged in the second virtual volume by writing the plurality of data fragments to a plurality of consecutive storage areas in the second virtual volume, and wherein after copying the plurality of data fragments to the second virtual volume, a copy destination logical volume close processing is performed, wherein the I/O processing unit is configured to: acquire a T pointer from a constitution table stored in a memory of the I/O processing unit upon receipt of a copy control instruction of an I/O request from the computer, the I/O request designating the copy destination logical volume which is to be a close target volume, the T pointer being a pointer that indicates a write destination address of a real area in the close target volume nearest to the copy destination logical volume, write the constitution table to the real area in the close target volume, the real area corresponding to the write destination address indicated by the T pointer acquired, and write the acquired T pointer, which includes the write destination address of the constitution table, to a header area in the close target volume.