You can use a backup or bootable clone if you have one. Hold alt (a.k.a option) at startup to select the other disk for booting.
NOTE: Time Machine backups are not bootable, you still need an install disk to 'restore'.
Without the system disks there is little you can do unless you have access to another Mac that has 10.6 installed?
Macs running 10.6 do not have 'recovery mode' that was introduced after that OS - so that is normal.
You will need to hit Ebay, Amazon or Apple.com to purchase the disks to stand a chance of examining or repairing the Mac.
http://www.apple.com/shop/product/MC573Z/A/mac-os-x-106-snow-leopard
The original disks have Apple hardware test included, that can sometimes highlight some hardware issues.
Another option is to try booting from another disk (like a USB drive), but you still need an OS X installer to get the OS installed to USB!
You can also try a genius appointment Apple store, but they may not help with such an old OS & Mac.