Repair Disk - Start up disk options

Solution 1:

One option would be to clone your startup drive to an external disk using something like SuperDuper! or Carbon Copy Cloner. Then you can use System Preferences->Startup Disk to select that external drive as the boot drive.

Once you've rebooted and are running the system off the external drive you can use Disk Utility to run the repair. After you're done, re-select the internal drive as the Startup Disk and reboot.

Solution 2:

One option that doesn't require any external drives or disks:

Disk Utility's repair disk is largely* a thin wrapper over the unix fsck (stands for "File System Check") utility. You can run it by:

1. Booting into "Single User Mode" by rebooting and holding command-S during startup.
2. A command-line input will appear; enter /sbin/fsck -fy
3. Wait for it to complete. If you see **** FILE SYSTEM WAS MODIFIED ***** then run it again, since sometimes fixing the first errors will uncover more.
4. Repeat until it says that the disk appears to be ok.
5. Enter Reboot to boot normally.

*I can't find any indication that Disk Utility's "Repair Disk" function does anything that fsck doesn't. Nonetheless, Apple recommends that you use Disk Utility instead when that is an option.