Obviously you've tried more than one charged main battery... Try also the hidden clock battery, which will be getting old now: https://www.ifixit.com/Guide/Canon+Eos+5d+Mark+II+Date+and+time+battery+Replacement/5012
It could be a faulty on switch as well. I had the on switch start to fail on my 5dC. I could get it to power on only if i wiggled the switch in the on position. It will still power down if my face accidentally brushes the power switch when I'm shooting.
