If the MacBook is not turning on without any response at all it's not the HDD/SSD that has failed. I'm sure Apple can fix whatever is wrong with it. The solution may be as simple as resetting the system management controller (SMC). If that should be the case, you can try it yourself:
- Plug in the MagSafe power adapter to a power source, connecting it to the Mac if its not already connected.
- On the built-in keyboard, press the (left side) Shift-Control-Option keys and the power button at the same time.
- Release all the keys and the power button at the same time.
- Press the power button to turn on the computer.
A scare like that should be all it takes to set up a TimeMachine backup
Cheers
Steffen.