MacBook – Time machine back up – how can I access the documents when the Mac has died

Try using the following Terminal command then restarting System Preferences (if it's open):

defaults write com.apple.systempreferences TMShowUnsupportedNetworkVolumes 1

I am unfamiliar with the two disk Mac Mini Server. I have, however, had a Time Capsule for several years.

First of all:

• Your file system is damaged beyond what Apple themselves can fix. Take the warnings very serious - you may loose data. I do not think that the hard disk is physically damaged, though.
• You may risk accidentially loosing your internal Time Machine backups by accident.

At this point in time, I would suggest that you first create an extra backup of your Time Machine data (this most likely require an extra external drive the size of your Time Machine harddisk) which for the Time Capsule can be triggered from inside Airport Tool - I do not know how to do it for your hardware.

That done, get a copy of your installation media as suggested by soxman. After ordering with AppleCare - due to the severeness of this problem - you may want to consider downloading a torrent with the dvd images so you can get started before you loose data.

When you have it fixed, consider upgrading to Mountain Lion Server (if possible on your machine). I do not know if the installer can reformat the file system and restore the time machine backup, but it might be a shortcut to the 10.6 start.

A final note: I would recommend against using third party file system tools for anything else than salvaging files before reformatting the partition. File systems are so important that I would only trust Apple itself for OS X file systems.