1. A method, comprising: a computing system establishing a logical address space for a storage device, and maintaining a map that associates logical addresses of the logical address space with physical storage locations of data stored on the storage device; the computing system allocating regions of the logical address space to respective virtual machines hosted within a virtualization environment of the computing system, wherein allocating a first region of the logical address space to a particular virtual machine comprises providing relative addresses to the particular virtual machine, each relative address corresponding to a respective logical address in the first region of the logical address space and being different from the corresponding logical address; and the computing system processing a storage request from the particular virtual machine, by: performing a first translation to determine a logical address for the storage request, the first translation based on a relative address provided to the particular virtual machine and an identifier associated with the particular virtual machine, performing a second translation to associate the determined logical address for the storage request in the first translation with a physical storage location on the storage device by use of the map, and processing the storage request at the storage device using the determined logical address for the storage request and the physical storage location associated with the determined logical address.