No single user account or group has full access to everything. You would need to change the permissions of all files and folders to add full access for this new group. Danger! Blowing away the standard permissions is a great way to cause difficult system problems; you shouldn't do that.
If you're trying to back up and restore files, you should use an actual backup program. Programs running as administrator (or as users that have been delegated the appropriate authority) can enable SeBackupPrivilege, which is an execution mode that allows reads to succeed regardless of ACL, and SeRestorePrivilege, which allows writing anywhere. Read more about privileges at TechNet.
If there are only a couple places you want to get into that only allow access to SYSTEM, you could have admins use PsExec to launch a program as SYSTEM. For example, this creates a command prompt running as SYSTEM:
psexec -s -i cmd.exe