How can I retrieve files from an old MacBook Pro that will not turn on?

If the MacBook won't turn on and cannot be connected in Target Disk Mode (https://support.apple.com/guide/mac-help/transfer-files-mac-computers-target-disk-mode-mchlp1443/mac) you may be able to remove the internal hard drive (depending on the age of the computer as newer Macs have SSD drives built in for which this will not work) and use a specialized cable to connect the hard drive to your Mac through USB.

Otherwise, the best you can do is to restore files from your backup. https://support.apple.com/en-us/HT203981