MackBook Pro, shots down by itself

It's failing one of the self-tests that run in safe mode. Usually that means the boot drive is bad, but it could be some other hardware fault. You could try running the Apple Hardware Test, but it's not very reliable.

Intel-based Macs: Using Apple Hardware Test

Back up all data to at least two different storage devices.

Boot into Recovery, launch Disk Utility, and erase the startup volume. This operation will destroy all data on the volume, so you had be better be sure of your backups. Quit Disk Utility and install OS X. If your Mac was upgraded from an older version of OS X, you’ll need the Apple ID and password you used to upgrade, so make a note of those before you begin.

When you reboot, you'll be prompted to go through the initial setup process. That’s when you transfer the data from one of your backups. For a walkthrough, see here:

Using Setup Assistant

Don't transfer the Guest account, if it was enabled on the old system.

Note: You need an always-on Ethernet or Wi-Fi connection to the Internet to use Recovery. It won’t work with USB or PPPoE modems, or with proxy servers, or with networks that require a certificate for authentication.

I don't actually expect that procedure to do any good, but you have to do it anyway to rule out anything other than a hardware fault. I think you do have a hardware fault.