If the laptop won't power on then you will have to rely on your existing backups. Some independent Apple Authorized Service Providers will back up your data for an extra fee if it is required to replace the drive during the course of the repair.
If the laptop powers on you can try putting the laptop into Target Disk Mode to connect it to another Mac to back up your files. You will need the appropriate cables & adapters.
If you have an older model laptop with a 2.5" SATA drive, then you could remove the drive and connect it to another Mac using a USB to SATA Adapter, dock, or enclosure.
Edit: Have you tried these troubleshooting tips?
SMC Reset:
https://support.apple.com/en-us/HT204652
Magsafe Power Adapter:
https://support.apple.com/en-us/HT203207
USB-C Power Adapter:
https://support.apple.com/en-us/HT204652