From what I understand your Vault Store was hosted on a 'storage system', you have previously archived data to this location.
Now somehow the Vault Store location on the Storage system has been deleted yet EV is still trying to access this location - This is the reason the services will not stay running.
You want to know now how to:-
1) Remove the Archives and items from within EV which were archived to the Vault Store which is now missing
2) Know how to get EV services to stay running
Well, doing it a bit backwards but:-
2) To get the services running you will need to recreate the folder which has been deleted from Storage, just the top level Partition folder should do for this.
When EV Storage Service starts it checks the the Vault Store Parittions, if even one parititon location (physical folder) is not available (missing or can not be reached for whatever reason) EV will stop the Storage and Task Controller services. Therefore you need to trick EV into thinking the Archived data is still there simply by creating the top level partition folder.
Dtrace StorageServer whilst attempting to restart the StorageService, this should tell you which Partition it is trying to reach and the path to that partition, recreate that path and start the service again, this time it should stay up.
For point 1, part of the deletion process is to try and delete the physical files, this obviously is not possible in your environment as they are already missing therefore EV will more than likely error on the deletion.
Your best bet is to do an EVSVR job against the missing Vault Store / Partition.
Run a Verify - Complete, check the log, it should come up with lots of errors, you will then need to run a Repair - DBReferences job against the same Vault Store / Partition but you will need to double check the Verify log before running a Repair.
To be honest, for point 1 you are better off opening a case with Support and we can talk you through it and ensure everything being done is done correctly.
**** I would not recommend carrying out an EVSVR repair job without a case open with support and after the technician has looked over the Verify log file and confirmed a Repair is needed ****