If you can't turn the ps3 on, a crossover cable would do no good, the ps3 would have to be powered on and accessing the internal hard drive in order to send it across the cable.
What you need is a laptop, or other device capable of reading a laptop hard drive. Removing the drive is not a warranty issue, so there is no problem as far as Sony still fixing it. I would borrow a laptop if I were you. Then set up the laptop with your pc... if you want to use a crossover cable, hook the cable between your pc and the laptop, (if your pc has only one LAN card, you'll have to take it offline) then go into your pc's network settings and temporarily configure it so your pc and laptop are part of the same network. Exactly what this involves depends on your OS and network, but if you have 2 LAN cards, it's usually called a "shared connection" in Windows. Once you've got it set up you'll have to enable file sharing, called SMB in windows and NFS in linux, then actually copying the files over will be the easy part.
Note: I'm not sure a computer can access the XMB partition, I seem to remember from somewhere that it's locked to everything but the ps3's native OS. Check the link below for definitive info on that. If it is, take your hard drive over to someone else's ps3, and put everything you can't replace onto a usb flash drive. There shouldn't be more than 2 to 4 gb of unreplaceable stuff. (everything you buy at ps store you get to download 5 times)
http://www.psubuntu.com
