How to encode overlapping videos?

[shampistols69]

I want to take a better quality 16:9 video and put it on top of a 4:3 video.
[Attachment 64194 - Click to enlarge]

[Selur]

You would open both clips in side your favorite NLE, use it's overlay/layering functionallity, adjust size and colors. (same can be done using an Avisynth/Vapousynth script)
You probably can find a tutorial video for most NLEs on youtube on how to overlay one video onto another.
For Avisynth: http://avisynth.nl/index.php/Overlay or http://avisynth.nl/index.php/OverlayPlus
For Vapoursynth: havsfunc, G41Fun, vsutils all come with overlay functions.

Encoding would be done the same way you would encode any video,...

[Cornucopia]

Mind the borders/edges, as the transition could be not just noticeable but jarring. You might consider an edge feathering to more smoothly blend the edge.

I note your reference has the 16:9 wholely contained within the 4:3. I hope you realize that for lots of productions, that is not how they masked the 2 different aspects - both had elements that were unique to that aspect. So re: your example, the 16:9 would be bigger, and have its left & right extend some beyond the edge of the 4:3. The result of combining the 2 doesn't give a rectangle, though, so some adjustment would be necessary (zoom & crop, prioritize 16:9, prioritize 4:3, fit all on larger rectangle canvas w background fill).

Scott