Hello and welcome to my User Tip
We get this question a lot, so I wrote this to assist and inform those of their options.
Your data recovery options
If your hard drive itself is mechanically dead (worst case)
Then a few thousand dollars can be spent for platter data recovery services to dissect and recover your data (provided Filevault wasn't used to encrypt the drive). Most users don't go this route, but it is available.
Before determining that a drive is non-functioning, it will have to be removed from the machine or external enclosure, tested and connected to other machines with special software that bypasses a lot to read the drive directly.
If the Mac or enclosure is dead, but the internal hard drive works
Then it can be physically removed from the machine and a SATA to USB adapter used to another Mac (or PC with MacDrive installed or a Linux machine) and the files recovered. Or special software used to directly control the drive to read the unencrypted bits.
Some Mac's the warranty, AppleCare or damage will occur if the drive is removed by non-certified Apple Repair folks, some will incur damage regardless as Apple likes to seal things up.
If the Mac is worthless as a functional machine, then a user can certainly use extreme methods to remove the storage drive out of it. Mac dissection videos can be found online at various Mac friendly repair and parts suppliers.
Migration Assistant can be used to migrate programs and files to a new machine if the Mac is dead, but the drive is fine using the SATA to USB adapter.
iFixit and Other World Computing
If the Mac and hard drive are fine, it's a software issue/cooprution that's causing you not to boot or access the drive
Depending what is causing the problem it can be fixed in place, leaving your files intact. OS X (in 10.6 and above) can be reinstalled in place, leaving almost everything else alone.
Yes, it is possible to recover data off a drive Disk Utility says it can't repair, or that OS X isn't booting from.
If the only way to fix your problem is a drive or OS X partition erasure is called for, before that occurs the Mac can be booted from OS X installed on a external drive, then either regular drag and drop copy methods or Data Rescue type software ($100) can be used to direct read the 1's and 0's of bits on the hard drive (ignores file structure etc) before erasure occurs. Data Rescue type software can also be used on the hard drive if it was removed from the machine using the SATA to USB adapter.
Try the drive on other machines, see if it shows up. Don't format the drive as even if it's not showing up, data can still be recovered if it's mechanically working and the file system is not.
These more extreme data recovery measures likely will not be undertaken by busy Apple staff, that's why your supposed to make backups before trouble occurs.
Local PC/Mac repair shops specialize in data recovery efforts which fall short of expensive dissection, which is usually handled out of town as it involves expensive equipment and the high cost/desperation by the original owner to pay for it makes for few customers.
Determine how bad the problem is
You can do a few things yourself to determine how severe your problem is and even fix it yourself.
One must determine if it's a easy self fix, a more complicated (but fixable by you or others) software problem or a hardware issue (you likely won't be able to fix yourself) and if it is the storage drive or not.
Try the drive on other machines, see if it shows up. Don't format the drive as even if it's not showing up, data can still be recovered if it's mechanically working.
Run through the Steps 1 - 8 and report the results of each step in your original thread, won't harm your data.
This will tell you right off how bad it is, by how far you get in the Steps.
Step by Step to fix your Mac
If you need to do data recovery
This link provides steps how to install OS X on a external boot drive and go about doing that and data recovery.
Create a data recovery/undelete external boot drive
If you have another Firewire capable Mac.
Consider your numerous data backup options later
Have a plan in place before trouble occurs.
Most commonly used backup methods
If this is all too much for you,
Consider a local PC/Mac repair as one of their main business is data recovery.
Then if your machine has a hardware problem, take it to Apple. Especially if it's under warranty/AppleCare, however they don't do data recovery and can just erase your whole drive or replace with a new one without your data.
If Apple says your machine needs a new hard drive, see about getting the old one for data recovery efforts elsewhere, or see if they will take the drive out of the machine, or will take the machine for repair if someone else removes the drive first.
Your data is usually of utmost importance, just don't give your only copy away, there are third party repair options for Mac's.
If your data is of the sensitive nature
Then you need to read this and determine your options to avoid others seeing your personal information
How do I securely delete data from the machine?