They need it immediately for a meeting? A meeting I assume they are going to be traveling to without their PC since the PC won't boot and they just need the data?
I know this is slightly cheating since you said "without taking the computer apart", but with PC's it is so easy and doesn't really require full open surgery like a mac to get the hard drive out.
Ok, I pop open the side panel in 10 seconds, disconnect the SATA cable and power cable from the hard drive in 15 more seconds, unscrew any screws holding the hard drive in place in the case in another minute, and insert this hard drive into an external enclosure with USB 2.0 cord. I plug the drive in to my PC at the workstation to quickly confirm that you can read the data off the hard drive with another minute of my time, then hand him the external enclosure w/ his drive in it for his meeting.
So, after 3-5 minutes he has his data and can take it with him in a very convenient way and can access it on any PC he wants.
Since you made this such a dire need issue, I gave the fastest solution. If it was not an immediate issue I would troubleshoot and repair the non-booting issue. You also didn't give any possible reason for why it wouldn't boot, so basically as long as the boot failure was caused by anything BUT the hard drive itself, this would work best (and on a Mac would be very hard since the hard drives are not easily taken out). If the hard drive itself was the issue, it wouldn't matter what method you used because the data would likely be corrupt and lost by then.
EDIT: I am sure that this is not what you wanted as my response, but since I did say that PCs had easier alternative methods for doing the same tasks, this was another chance to show you just how that might be the case. PCs are so easy to take parts out and put new parts in, so it really erases the need for TDM to be used in data recovery for the issue you just described. Thus proving my point I've been making this whole time.
