Migration assistant to specific folder
Migration assistant will copy files into expected places on the disk so ~/Library ~/Documents etc and is only meant to be used when installing OSX.

If you want to copy to one directory don't use Migration Assistant. Instead 


Install OSX from scratch and create a new user (for ease if you have multiple users create them in the same order as before so that their UIDs match) - or just keep the target running current OSX.
Connect Macs via firewire and boot up old Mac in Target Disk Mode 
The old disk is now seen under /Volumes
Copy files onto new disk as you want


If step 2 fails then you need to take the disk out of the old Mac and put iy in a USB or other external disk caddy. Then it might work. If the disk is corrupt then look at disk recovery tools or firms.