Upgrade os snow leopard to mountain lion without time machine
Rule: Always back up at least your own document, photos etc — or your whole User Directory, if you've got the space - before any big changes. 

You don't necessarily need to backup the OS — though a disk clone onto an external USB, made with SuperDuper or CarbonCopyCloner, is an instant rescue, right at hand, in the event of a crash or data loss, as well as another Startup Disk from which to repair your hard disk. 

If you've got access to an OS install image, download or disks, in the event the upgrade doesn't go well and you need to downgrade, you don't need a clone (though it's easier) but just a copy of whatever you don't want to lose. 

Related Rule: Always routinely back up / copy to somewhere else anything you don't want to lose. 

You don't need Time Machine. It's just one method, one kind of external drive. 

I use Arc, for example, which backs up what I tell it to (to very cheap storage using A3) every night, silently, from the menubar. I also clone every few weeks to a small USB external.

Another inexpensive way is to copy / backup to a flash stick - mine holds 16GB, which is enough for my work. A photographer or video person would of course would need much larger backup and storage.

It's a broad subject — but just don't mess with big changes "unprotected." Data loss happens. 

Save your data to DropBox, that's one easy step.