How can you back up os 10.5.8?

Time Machine backsup the drive to an external drive, if that is what you are asking. Also, Carbon Copy Cloner, free trial that is fully capable, will make a bootable copy of your system.

I don't understand why an external drive would require 10.6...the Mac OS X format (HFS extended Journaled) and partition scheme, GUID, have been used since Mac OS X 10.4. The Time Machine system has been in use since 10.5. Someone may be giving you some bad information.

Check the drives offered by OWC, http://www.macsales.com and see what they have. I have three of their Mercury Elite pro external drives, 2 TB each, and they are great drives. OWC has on-line and telephone help so can answer any questions about compatibility of the drive you are interested in.

An external drive can be partitioned into a partition for Time Machine to use for backup, or another backup procedure, and a work space partition. I setup my drives, for three different computers, to have a backup partition and a Carbon Copy Cloner bootable partition and a storage partition. That way there is a good backup of the system, and a bootable copy of the system so if anything goes wrong I can simply boot (restart holding the Opt key then select that partition for startup) into a working version of the operating system.

All the external drives that I've seen require 10.6

Maybe this refers to some software that comes with the drive, which can be ignored. A hard drive's adaptibility is determined by its format. If it is formatted correctly by you using Disk Utility, it will work on your 10.5 Leopard machine, no problem.

You're wlecome and good luck with the backing up.