you can download some live-cd like Knoppix, boot your pc with it and then burn your data to backup dvds or cds...you'll need Knoppix for this (can be downloaded from web, burn on your friend's if you can) and a cd/dvd writeable drive...I guess this is one nice solution. another is, take your HD, move it to another PC with linux installed, mount it and copy/burn files to some backup location..
EDIT: or then just (re)compile a kernel that suits your new mobo and cpu...I guess you can do this with another machine or with a live-cd (like Knoppix), if you jus know what hardware you got
grab the sources from kernel.org and if you have, copy the .config -file from your older sources dir, or if you don't have it, then just compile a new kernel. then, using Knoppix or some other live-cd, add it to your bootloader or replace the older kernel (I recommend not to overwrite) and boot.