We often see customers with production systems utilizing Exadata. Because Delphix uses Oracle API's to sync with production, there is no difficulty there. Delphix is logically like a backup destination.
For a Virtual Database (VDB), the database server continues to run the Oracle binaries and create an SGA and manage user connections. There is no logical difference in the "instance level" config of a VDB using Delphix compared to a database using physical storage. Typically, servers hosting VDBs have minimal storage--just enough to host the OS and database binaries (~50-100GB).
Each Delphix Engine has it's own storage, typically using enterprise class SAN vendors like Hitachi and EMC. One can choose any storage which VMWare or Amazon EC2 supports, from a single lowly drive to a pure SSD array. The key is to get the required IOPS and storage footprint. To a VDB, the Delphix Engine looks similar to NAS.
Using Exadata for the Delphix Engine storage would not make sense, as the features which make Exadata worth the money could not be leveraged by IO running through the Delphix Engine. The Delphix Engine does not run Oracle (or any other vendor database software). Other storage solutions from Oracle/SUN could be used for Delphix Engine storage, provided the required performance characteristics are met and they are supported by the hypervisor.