You'll need to repair your OS X installation. You can do this without destroying data on your Mac. This post from Apple Discussions explains it well:
Do the following:
- Repair the Hard Drive and Permissions
Boot from your Snow Leopard Installer disc. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities menu. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the installer.
If DU reports errors it cannot fix, then you will need Disk Warrior and/or Tech Tool Pro to repair the drive. If you don't have either of them or if neither of them can fix the drive, then you will need to reformat the drive and reinstall OS X.
- Reinstall Snow Leopard
If the drive is OK then quit DU and return to the installer. Proceed with reinstalling OS X. Note that the Snow Leopard installer will not erase your drive or disturb your files. After installing a fresh copy of OS X the installer will move your Home folder, third-party applications, support items, and network preferences into the newly installed system.
These instructions were for Snow Leopard, but the same formula will apply for Mavericks. You'll need bootable installation media--Mavericks doesn't come on CD/DVD, so you'll need to make bootable installation media, but examples of how to do this are online, including official instructions from Apple.