Recovering mysql database from an unbootable hard disk

We have installed a Linux Authentication Gateway (run on Ubuntu Server) for more than a year ago. It has several features built in upon install such as Squid, php5, mysql, etc.

The thing is this: we see that the OS might have some corrupted crucial data structures. The system is now no longer usable. For instance, upon booting up, it complains a lot of "segmentation fault" and several commands have [fail] instead of [ok]. The list is just huge and let's just say that we can't do anything from there any more. We can't even log in. When you log in as root, the OS simply bounces you back, asking you for your username again and again, in an infinite loop.

So, now, we want to at least recover our mandatory files. We basically use a live CD, boot it up and see things in our hard disk. But the mandatory directory: "mysql" located at var/lib/ is inaccessible. On the window GUI, the directory is visible but it has a rectangle with an X on it. When trying to access, copy or compress the directory, a message box popped up and complained that we do not have the privilege to do such action. Sounds like a privilege problem? Well, we simply turn on the terminal and navigate to the var/lib/ directory, hoping to do some simple chmod. However, what is shocking is the mysql directory is INVISIBLE. ls command does not list it out. Using some directory check bash command, the program says that there is no mysql directory to be found as well. And all these commands are already under root privilege.

So in this problem, we want to retrieve everything in the mysql directory via Live CD. How do we accomplish this?