Reformat and then use Time Machine? I want to reformat my MacBook late-2008 unibody aluminium with HDD.

Can I do that and then after I have reinstalled OS X go to Time Machine and recover everything? Is this possible?
Yes, this is possible. Make sure that you've completed a backup through Time Machine, and that the backup is working and accessible, then erase and install OS X.

Once OS X is installed, you can use Migration Assistant to restore your data, or set up as a new Mac and use Time Machine to restore individual files/folders.