Without Sandboxie, is there a way to run multiple copies of Steam?

This is not possible without using Sandboxie or a virtual machine.

This is because the Steam client service (the core of Steam) runs as System and has access to all user directories. You can set the service to run under a specific user instead of Local System, but that will prevent other users from running it.

There is nothing you can do about it. Steam is intended to work as a singleton application.

One reason might be because some games save their player-specific data in predefined folders, rather than in %user%. Such games would still work when running Steam in Sandboxie, as there would be no contention between the instances, since they will be using different files.