The answer depends on just how broken your HDD is. The HDD could be unbootable because of some bad sectors or because the partition table is screwed up. In that case you might be able to recover some or all of your data using data recovery software.
If that doesn't work, you could try to mount your HDD in an external drive enclosure, plug it into another computer and see if the HDD can be read by the other computer. If so, you're in luck. Just copy your data over to the other computer.
If the drive is physically damaged, say from a head crash or a bearing failure, then no software in the world will be able to retrieve the data from that disk. There are companies that specialize in recovering bytes from disks that are physically damaged (but not completely destroyed). Their services are not cheap -- it could cost you over $1K without any guarantee it'll work. I doubt that your data is worth that much, because otherwise you would have backed it up before the HDD failed, right? ;D