How to fix my MacBook Pro that goes straight into sleep/hibernate?
Before doing an SMC Reset as suggested by @Thecafremo, I decided to clear the PRAM/NVRAM (because the SMC Reset page says: "an SMC reset should only be attempted after all other standard troubleshooting has been performed")

This seems to have been enough to bring the MacBook back to life at least for the moment.

Clearing the PRAM/NVRAM is achieved by holding down Command+Option+P+R during boot before the gray screen appears and letting the startup go thru three chimes before releasing the keys.