If the laptop does boot up normally without the battery and is just connected directly to the AC adapter, that could only mean that the battery needs to replaced already. Do also try a different AC adapter that has the same voltage and see if the same problem will persist or not. If you're getting an error about the HDD, unfortunately I would suggest you proceed with the back up just to be sure.
Here are some other troubleshooting steps that may help if ever the laptop will not boot up.
- Start by doing a hard reboot, remove the battery and unplug the AC adapter then press and hold the power button for 20 seconds then try booting it up again.
- If this will not work, do test the AC adapter if it's working properly. Remove the battery and power the laptop with just the AC adapter connected and see if it will boot up or not. do also try a different AC adapter that has the same voltage and see if it will work or not.
- Proceed with reseating the RAM's if trying a different AC adapter will not work, remove them all for couple of seconds then put them back in making sure everything is seated properly.
- If it will still not work, connect an external monitor and see if you'll get any display from it. If there's a display on the external monitor that could mean that the problem is the built in display.