Backup without using TimeMachine?

emails are never included in a backup; they are saved on the email server.

Your post is not entirely clear. It sounds as if you're saying you have already lost the data. If there is no backup, the data is gone.

If you want to back up your Mac without using Time Machine you can certainly do that. You can manually move everything you want to save to an external drive. Or you can use software like CarbonCopyCloner. Note that backing up now will not recover any data already lost.

If your data are gone, they may already be gone. But alternatives to Time Machine are things like SuperDuper! or Carbon Copy Cloner. Both of them create complete clones of the internal drive onto an external drive, if that’s all you need to do the free version will work fine (the paid version typically does incremental back ups scheduled back ups, etc.).

<< there was also a problem from my email-provider, I had still an old pop account, >>

That's not a problem with your provider, that is operator error -- Yours.

POP emails are indeed stored on your computer. But you should NOT be using POP today, because it can not handle multiple readers (e.g., iPhone AND Mac).

When using IMAP, as you should be, the emails are safely stored on upper providers server, and are not sent you until you choose one to reading in its entirety.

Time Machine's biggest "claim to fame" is that it is the backup that gets done. It does not ruin performance of the rest of the computer while doing its backup operations. You do not have to set aside a "Special Time" when you only do backups. When you need it, your Time Machine Backup is much more likely to be there.

Another vote for Carbon Copy Cloner...

http://www.bombich.com/software/ccc.html

See Legacy backup if you want the whole backup.