A data storage system has a primary data storage subsystem including primary data storage and a secondary data storage subsystem including secondary data storage. The secondary data storage contains backup versions of data stored in the primary data storage. To permit a host processor to access a specified backup version, the primary data storage subsystem assigns a virtual storage address to a specified backup version, and the host processor sends to the primary data storage subsystem data access requests that specify the virtual storage address. If the primary data storage subsystem has spare data storage, then a copy of the specified backup version is read from the secondary storage and written into the spare storage, and the address of the copy in the spare storage is mapped to the virtual storage address and accessed in response to the storage access requests from the host processor. Otherwise, a block-level direct access of the secondary storage is performed to access the specified backup version. In either case, the backup version appears to be in the primary data storage subsystem when the host accesses it.