A RAW sector-by-sector copy means that each and every sector of your HDD needs to be copied. Being a RAW sector-by-sector procedure (whether direct copy into another drive or copying to an image file) means that the content of each sector has no meaning in terms of being a certain file, or being "empty", or being part of a certain partition or filesystem. To put it "in binary terms", where there is a "zero", the copy gets a "zero"; and where there is a "one" in the original, the copy will get a "one" (except if the original sector can't be correctly read, for whichever reason). The copy procedure (and the program you used, which you didn't mention) doesn't care of the meaning of those zeroes and ones.
OTOH, instead of using a RAW sector-by-sector copy, you could use some other "more intelligent" (in a certain sense) program that indeed understands about the specific filesystem you might be using, and where the partitions start and finish (and every other meaning of those zeroes and ones). Then the resulting destination drive is not an exact RAW sector-by-sector copy of the source HDD, but the files are indeed the same (or at least the copy / backup program will try to make an exact copy of the files).
UBCD and PartedMagic (among others) have several of these type of programs.
Alternatively, you could try accessing the problematic HDD by means of PartedMagic (or by means of the copy program you choose to perform the task), and then select the specific files / folders you are interested in and copy those to your external drive.
Part of the programs may try to copy a file and if / when the reading would fail, they move on. Some other programs are more about rescuing the data (maybe sector-by-sector, maybe understanding the filesystem in the source HDD you want to recover, depending on the program).
The method to be used (from the 3 I just mentioned) depends on what you would be trying to achieve. Apparently, the first method (RAW sector-by-sector) is not so adequate for the hardware available to you ATM. So either you need to select some other method / program, or get some adequate HDD (equal or bigger than the original).