Many Linux distributions include an easy tool to make bootable "Live" USB flash drives (use a small 3-8GB USB), including Fedora, Ubuntu, Puppy Linux, Mint, etc.
Once you create a live USB, you have to set the BIOS to boot from USB first (may have to disable UEFI boot on some machines to allow USB to boot first).. Once you boot to Linux, you should be able to see the existing HDD, and you can plugin other external backup USB drives for copying files, they should be mounted automatically. You may have some minor issues with drivers on some Linux distros, but modern ones should work without a hitch.
I recently had to do that for some laptop (proprietary Sony SSD without a SATA connector, grr).
Alternatively, you can just pull the suspect drive and connect it to another PC to copy/backup files.
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