If your Mini is about 1-1/4" tall, you have the 2010 model, if it's about 2-1/4" tall, it's 2009 or earlier. For the 2010 models (but not earlier), if the updates to the OS were applied, even if it's 10.6.X, you should be be able to run Internet Recover by pressing Command-R after turning the computer on. This requires network connectivity. It could take a while to get it loaded. Once there, one of the options is Disk Utility and there should be the option to eject the optical disk (if the steps suggested by the other posters doesn't work). Also, one of the reasons why Disk Utility may not have been able to do some repairs is because it can't do certain repairs on the disk that the OS is running on while it's running on that disk. If you can run Disk Utility that (hopefully) loads with Internet Recovery, it may be able to complete the repairs (it also may be that Disk Warrior did the same type of repair that Disk Repair tried to do). There are other options to run the OS off of the computer but from another disk (if you use other Macs).
As one of the other posters suggested, it may be time to replace the hard drive or upgrade to a SSD. There are various issues involved so when you get to that point, feel free to post again.