Assuming your hard drive is functioning properly, you can use it as an external startup disk.

This applies if your drive is an external enclosure, or if your Mac is in target disk mode depending on the model and versions of macOS—newer versions of either may disable this option.

On the working Mac your disk or device is connected to, press and hold the Option key as you power on. This should bring you to the startup disk selection menu.

In the list of startup disk options select the connected disk.

Booting may take much longer with this method due to lower available bandwidth with this sort of connection.

Once booted, you can use Time Machine to make a full backup of your device as though it were functioning normally.