Q: Hi. My mavericks is crashed and when i on my mac always showstha disk utility window. I do everything like repair and all but not fix this problem. I have no back up of my files. Is there any way to reinstall mavericks without erase my old data ?

This Apple article explains how to boot your Mac into OS X Recovery and reinstall the operating system. That re-installation should not affect your other files on the disk. But you should still make a backup of your boot drive. If you connect and properly format an external drive of sufficient size, you can use Disk Utility > Restore to make that backup.

To back up your data, either use Target Disk Mode and a cable from another Mac (which mounts your problem Mac as an external disk of the other Mac) and Disk Utility on that other Mac, or — and this is all assuming you either have a slow Internet connection and don't want to use Internet Recovery, or if you have an older Mac that doesn't support Internet Recovery — you can create a bootable USB key disk on another Mac and then boot your problematic Mac from that keydisk, then use Disk Utility from the installer to make a backup to another (external) (scratch) disk.  You can also use that keydisk to reinstall OS X, to repair or to reinstall OS X.