Disk utility says hard disk needs to be repaired, button doesn't allow

Note: Backup before doing anything that may cause loss of your data. if you don't have one yet.

Seeing as this may be a bad OS install and the repair from disk utility is failing. Do you have a backup of your data? If you do have a full backup you might want to try a clean install of OSX Mavericks using a bootable USB, this will require another mac computer to create or your own computer can do this as well if it still boots. To run a clean install, when you boot into the installer you will have to manually reformat the drive using the installer disk utility and then run the mavericks installer.

OSX Mavericks bootable usb creation guide:

http://howto.cnet.com/8301-11310_39-5760...

Also if you have the original CD's that came with the computer you can reinstall the original OS that the computer came with and then upgrade it to mavericks off of a fresh install. To boot to a CD on the mac first insert the install CD and then hold down the 'C' key on startup. Just noticed this I think you may have done this already but the post was a little unclear.

NOTE: IF YOU DON"T HAVE A BACKUP, BACKUP FIRST!

Since the computer seems to still boot try retrieving and backing up your data like you normally would.

If it doesn't boot:

Second to retrieve data you can try to boot the computer into target disk mode by pressing the 'T' key on boot up and then you can connect the mac to another computer with a firewire 800 cable in order to pull the data off. Again this may require another mac with a firewire 800 port. If this is not an option and adapter is not a bad way to go either and they don't often cost to much, they can also be purchased and a store like Fry's or Best Buy.

If you do not have a full backup there are other ways of retrieving the data from the drive itself before you reformat to fix the issue. You can use adapters like these: http://www.amazon.com/Vantec-CB-ISATAU2-... . these adapters are all different prices so you may want to look around the one I posted was an example. There are also external enclosures that can allow you to put your HDD in them and use it as a USB Hard drive to retrieve data.

Repair guide to retrieve hard drive from your mac mini:

Mac mini Model A1283 Hard Drive Replacement

after this you can try a reformat and then install the original OS using the original CD's or install mavericks with a bootable USB. If this still doesn't work your drive could be physically damaged at this point in which case it is a small cost to get a replacement drive.

I believe this covers everything, Good Luck.