Is reinstalling Mavericks possible via recovery mode with two HDs and without a backup? I can't get my iMac working and I don't have a backup.

If I choose to reinstall Mavericks or Lion (it shipped with Lion, I upgraded to Mavericks) will that wipe over the core HD?

I got core HD 350 GB and another internal HD of 2 TB.

If I choose the install destination to be the 350 core HD, will that wipe out data on it?
If you reinstall OS X from Recovery HD or the installer app (or Internet Recovery), your Mac's HD will not get erased.  If the installer needs to erase things, it will tell you, so go slowly and really read the words if you get a dialog box and choices. 

If you are concerned, install OS X onto any free USB drive and then get a good backup of the system you care not to loose.