OK after a fair amount of Googling and trail and error I found a solution. I've written up what I did below to help other people in the same situation.
Firstly you have to get the hard drive out of the device this is harder than it sounds as the device is full of clips keeping it shut. Fortunately there are some disassembly images here:
I removed the fan first using the screw on top, and sliding it out from the side. You need to reach inside and unplug it as you pull it out. Next, look in the gap where the fan came from and you will see a clip on the inside of the fan vent. Unclip this.
Looking down on the unit you will see three small holes, using a pin push through these undoing the clips on the inside.
On the underside of the unit there are some stickers, peel these off and you will find two more holes to access clips through. Once these are undone you should be able to pry the unit apart, it still takes a bit of force.
Once inside you should have a circuit board with a hard drive attached. The hard drive is attached through one side screw and two screws underneath. Remove all of them. You should then be able to slide the drive out.
The drive is a 3.5" SATA Samsung HD502IJ. You can put this into a SATA caddy to attach to a PC.
Once attached to a PC running Windows 7 the PC will install the drivers for the device but will not show the drive in My Computer. If you go to manage your computer and then disk management you will see it shows a number of partitions on the drive but shows them as blank. Don't despair. I found the reason this happened was that it is not using a standard Windows file system. You need to find some software to read SGI XFS file system.
I searched around and found UFS Explorer:
You can get a shareware version and try it out on the drive (you do need to pay for it to transfer any files over 64kb off it though). I tested the drive with the shareware version and then bought the full version to get my files off.
I managed to copy off most of my files using this application. On the downside I did have one folder with lots of files in and UFS Explorer doesn't appear to be able to read them all. I'm not sure if this is a deficiency in the software or a weird way the LinkStation has stored it's data. Either way I've lost a few things but not everything.
I'm going to now buy a RAIDed NAS to avoid this in the future.