I'm looking for a way to turn an HD video clip into 80's style VHS low-quality. Here is an example of the look I'm desperately looking for:
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Every decent video editor has built-in filters (like Color Correction, Film Effects, TV simulator, Color Curves, etc.) that can be used to achieve the desired 'look'. There are plug-ins and tools that contain many ready-to-use presets, such as:
This is certainly more work, but if you want a truly authentic and "organic" VHS look:
Digitizing analog video is a little more complicated, but really any TV tuner card will do. Your graphics card may already have RCA composite or S-video component input. Or if you have a set-top DVD recorder, then you can "digitize" it that way, and then rip the MPEG2 video from the DVD on your computer using DVD Decrypter (a fantastic tool and completely legal if you're using it for video that you own the rights to).
Check out VirtualDub for a good general video capture and processing tool to help you with this.
Besides the plug-ins and preset looks (which are great and life saving most of the time) you can manually create such looks by observing some example videos. The video you showed, for example, has some blur, and overall diffuse. It has plenty of colour noise. It is pretty brightened up, with lower contrast. There's a visible shift in color channels. It has a reddish and yellowish look etc..
Also you can add some bad VHS tape artefacts as mentioned here.
Of course creating each property manually might need a capable software and some experience with the software you are using.