There's a couple of of hurdles here.
The first hurdle is, what about the original HD is causing the PB to shut off like that? That's interesting behavior for just a failed HD; usually the computer will either refuse to boot from it (flashing "?" while it looks for a bootable OS) or will get hung during the boot process. For it to shut off because of the HD isn't common in my experience.
Assuming the original HD is still accessible -- it spins up, the computer can see its SCSI ID, and all the files are there; it's just not bootable -- you should be able to retrieve the data from it. The HD in the PB140 is SCSI (and a 2.5" drive as well) so under normal circumstances you'll have some difficulties in getting a cost-effective solution. HOWEVER, you said that you bought a PowerBook 100 as well. If that PB100 is still functional then you're mostly home-free as the PB100 supports SCSI Disk Mode. Install the PB140's hard drive into the PB100, connect a special (NOT REGULAR) SCSI cable to both PBs, turn on the PB100 first, wait for the SCSI icon to appear on the PB100 screen, then turn on the PB140 and let it boot (this assumes the PB140 is bootable from that other hard drive). Once you reach the desktop, the hard drive in the PB100 should appear on the desktop as a regular SCSI hard drive. Copy whatever file(s) you want from there and you're done.
The biggest issues with SCSI Disk Mode is (a) finding a working SCSI Disk Mode adapter or cable, and (b) the assumption that the data on the original hard drive is still accessible (i.e., not a drive failure). If the drive is spinning but otherwise inaccessible your next step would be to find an identical (IDENTICAL) hard drive mechanism and carefully swap the controller boards between them.
I don't envy you this adventure. :-)