When you say power supply, you are talking about the power strip that everything gets plugged into, correct?
Not quite
Inside the case itself is a box where the power gets input from the wall, and looks something like this:
The power supply converts the 120V or 240V (depending on where you live in the world, of course) to the 12V, 5V and 3.3V portions the computer needs.
If the 12V is fine, the Hard drives should still function normally, but if the 3.3V is faulty the DVD/CD drive may struggle and or cause a short.
If possible, try a friends DVD reader/writer to cross out the issue of your own optical drive.
If it isn't that, it is definitely pointing to a power supply issue.