You should be able to back up your data by booting from a cd and accessing your hard drive.  I'm not sure what resources you have and I'm not at all familiar with mandrake but with redhat, after you answer the first or second question when booting from the install disk you can alt-fx to get a root terminal session from which you can mount your hd and copy/move your cd elsewhere.  In one case where I did this I was able to tar my home directory and ftp that to another box.

Very likely you can reinstall mandrake and tell it not to reformat your filesystems so that when you come up your home directory will still be there.  We'll have to wait for someone with mdk experience to verify that, but I'd be very surprised if that option is not there.  It's likely to be called an "upgrade install" or some such.

As long as the hd root sector has not been touched (which would mess up partition tables) and the fs's have not been formatted your data is still there and accessable.