How to extract a filesystem from a disk image

You need to backup an entire hard disk to a single file. Supposing your disk is at /dev/hda and the backup file is image-file, you’d do:
# cat /dev/hda > image-file
or
# dd if=/dev/hda of=image-file

The file backup you get will hold a copy of every single bit from the hard disk. This means that you also have a copy of the MBR in the first 512 bytes of the file.
Because of this, you can see the partition table on the backup file:

# sfdisk -l -uS image-file
Disk image-file: 0 cylinders, 0 heads, 0 sectors/track
Warning: The partition table looks like it was made
for C/H/S=*/255/32 (instead of 0/0/0).
For this listing I'll assume that geometry.
Units = sectors of 512 bytes, counting from 0
Device Boot Start End #sectors Id System
image-filep1 32 261119 261088 83 Linux
image-filep2 261120 4267679 4006560 82 Linux swap / Solaris
image-filep3 4267680 142253279 137985600 83 Linux
image-filep4 0 - 0 0 Empty

Now, suppose you want to extract partition number 3. You can see that it starts at block 4267680 and is 137985600 blocks long. This translates into:

# dd if=image-file of=partition3-file skip=4267680 count=137985600

Now, peeking into the contents of the partition is as easy as:
# mount -t ext3 -o loop partition3-file /mnt/hack