I used /Applications/Utilities/Disk Utility.app to check root partitions for problems and Verify Disk function said there were some issues.
But there seems to be no way to run Repair Disk - it wants me to run OSX installation disk and select such option from there.
The problem is - DVD drive doesn't seem to be working here. I haven't used it in years, so I had no clue.
Any ideas how I can repair disk?
If you boot OS X with the Shift key held down, it'll start in safe mode and (among other things) run a repair on the startup volume as part of the boot process (note that you'll then want to reboot normally, because some things are disabled in safe mode).
Another option is to boot with Command-S held down, which starts in single-user mode (very minimal environment, just a full-screen command-line interface); you can then run fsck -yf to repair the volume (note that fsck doesn't always fix everything on the first try, so if it repairs anything you should re-run it until it says the volume is ok).
Personally, I prefer the single-user option because it'll tell me about what it's repairing (safe boot just shows a progress thermometer, with no indication of what it's finding and/or fixing). For more details on both options, see Apple's KB article #TS1417.
In 10.7 and later versions you can repair the starup drive from the recovery partition. Hold ⌘R on startup, select Disk Utility, and press the repair disk button.
