That means the Mac van find a startup drive.
If you have Lion or later boot into recovery
and go to Disk Utily and repair/verify disk.
Otherwise you will need to boot from an install DVD
What model Mini do you have and what OSX version was on the Mini?
The 2011 Mini and later do not use installation media. Then can boot for interent recovery even without a HD
the firmware for the 2010 Mini (which has optical drive) could be update to provide the same capability
The 2009 and earlier ' boot from the internet and need optical or USB media to boot/install to a blank HD. If you Mini came with an optical drive you can call Apple and order install media for a price.
My mach mini is 2009 and earlier, I entered in the serial on the bottom to instant chat but it says its not valid, which is another issue I can talk to them about.
I believe the optical drive you are referencing is the DVD Disc drive in the front of it? If so than the last fix is the last resort.
Can I tether my mini to my non apple laptop to install updated software?