Wont Power On
No matter what you do, you can't get your device to turn on.
If your device won't turn on try charging the battery. If that does not work you may need to replace the battery. There is a replacement guide here that details how to replace the battery.
It is possible that it appears nothing is happening because the display is bad. If you can see the power light come on and nothing happens this might be the case and you will need to replace the display.
Audio from the device sounds distorted.
Loose Headphone Jack
Test audio through the computer speakers, then test it using headphones. If the audio is only distorted while listening via headphones, the audio jack or the headphones themselves may be faulty. Audio that cuts in and out is often a sign of a loose audio jack, so make sure the audio jack is fully inserted in the computer.
If the sound is only distorted through the speakers they might need to be replaced. A replacement guide can be found here. First make sure the speakers are connected by checking the cord and reconnect them to the laptop if necessary.
Sticky or Missing Keys
Keys either are missing or wont respond properly.
No Keys Respond
If none of the keys respond, the keyboard may be broken and needs to be replaced. Before then, test to see if it is still properly connected in the device.
Keys Can't Be Pressed
If a key can't be pressed or it experiences resistance, there may be something below the physical key holding it up. To fix this, remove the key and make sure nothing is stuck below it. Once it is sufficiently clean, snap the key back into position.
Keys Are Missing From the Keyboard
If keys on the keyboard are missing, replace them. There is a replacement guide here.
The device keeps overheating when on.
Dust in Vents or Fans
Most likely the fan will have taken in dust and hair over time. It will eventually become slower or unresponsive. In that case use a compressed air duster to blow out the vents. If the problem persists, you may need to take off the back panel of the computer and clean them manually.
Out Of Storage Space
The device wont store any more files.
It is probably best before you replace the hard drive to go through the files that are saved on the device and remove ones that are not needed. Since the chromebook is integrated with google drive, you can move many of your documents there. Or move some files that aren't needed regularly to an external device such as a flash drive or SD card.