Warning: I've never tried this so I don't know if it will work with ESRI services nor if this will cause black ESRI helicopters to raid your site ;-). I have done it for other services and I know it can work but I make no warranties. I welcome other more knowledgeable ESRI users to share their experience.
That being said, let's get into the details.
@Thomas who answered before me is right: mapped network drives are not accessibile to Windows services.
But on the other hand Windows services can be setup to access the UNC path to which the drive is mapped.
For instance: drive Z: is mapped to \\someserver\someshare
A windows service can't reference Z: because it doesn't have a logon session so no drive mappings are possible or recognized. But the service can still access \\someserver\someshare if the service runs under a user account that can access the share.
Usually this is achieved by going into the Services control panel and changing the user account under which the service logs on. Typically this is LocalSystem and you would then change it to something like DOMAIN\USER (or SOMESERVER\SOMEUSER) where DOMAIN\USER is an account that has access to \\someserver\someshare.
If the ESRI service is already started as a regular user (ie not LocalSystem) then things are much easier. Simply create a user with the same username and password on someserver and assign the appropriate privileges so that it can access someshare.