If you use a Linux live-CD, you can use
ddrescue to create an image of the whole HDD and store this (probably large) file anywhere you like.
ddrescue is similar to
dd in the way that it will create a raw disk image. But it is designed specifically to work with slightly broken HDDs.
Many websites like ForensicsWiki suggest multiple passes of
ddrescue to get fast and satisfying results (you need to use the same logfile every time, so it can know which sectors were already tried):
ddrescue -n /dev/sdX NETWORK/diskimage.img logfile.txt
ddrescue -r4 /dev/sdX NETWORK/diskimage.img logfile.txt
(For best results, you might want to have a good look at the manpage or other tutorials using ddrescue, depending on the damage on your HDD)
You can then mount this image (using the
mount command from Linux/Cygwin or tools like OSFMount for Windows) so it is accessible like any other folder structure.
There is one caveat to your solution using the image: if you have a raw image, the empty space in the filesystem is also copied (so a 500GB HDD that is only filled with 300GB data will still have a 500GB image to copy over the network).
However, if you have enough local storage, you could first keep the image there and then resize it using tools like