turns off on startup after update

Using the App Store Updates, I recently (tried) to install an update (possibly a recent security update). A restart was required. When it tries to start I get a progress bar on the gray screen. About 1/2 way through the computer turns off.

The computer is a desktop, and the hard drive is almost full.

(My laptop's HDD died, and I haven't got it fixed yet. Once I do, I can delete some files from the desktop to create room.)

Thanks for any ideas.

Beth