How do I extract data from HD on mac that won't boot?

A couple days ago I got the flashing question mark folder at start up. I tried to boot in recovery mode but no disk shows up in start up disk utility. I'm hoping I just need to replace the cable inside the macbook.

But more importantly - I need to get the data off that hard drive before I go around trying to fix this computer. What is the best way to get the data off hard drive safely?

1) Should I get an external hard drive and copy data from recovery mode? Is this possible, and if so how do I do that?

2) Should I remove the hard drive and use an external enclosure and use a different computer to copy that data to a second external hard drive? Keep in mind I don't have access to another Mac so it would have to be done on PC.

Any and all advice would be appreciated, I really need that data! (and I really need to get this computer to turn on!)