How-to, easy method. (Does not work if there is anything, such as a logo, in the bar area)
Open the video in Handbrake.
Pick the preset that best matches the source (or what you need to convert it to). I recommend setting the frame rate to "same as source". Also be sure to "add all" in the subtitle tab to preserve captions.
The letterbox part - In the dimensions tab, just pick "automatic" under cropping.

If there IS a logo in the bar area, you'll have to use manual cropping.

Don't be alarmed at the smaller resolution. The "resolution" you lost was just black bar.

Note: for those of you attempting to use retro equipment that uses analog captions, this will strip off the analog closed captioning data. However, I don't think that data is usably preserved anyway in digital video.
This will also strip off anything else that is written in the blanking area - so keep that in mind for whatever use cases I can't think of.

I know video had some interesting shit back in the VHS days. I have no idea if any of those interesting things still work storing it as digital video, but if it can be saved, letterbox stripping will definitely destroy it.

If you don't know what this is talking about, normal movies and TV only have caption data there and in a method that will not work on modern playback devices any

@hellomiakoda the extra info is written in lines that don't show up on the screen (because the cathode ray is switched off then) so they don't get preserved in the conversion

@noiob This may depend on how it's captured. For example, DVDs used to output analog, so ripping the DVD MAY have that data in the resulting file.
Also, I have myself captured over scan video, which may have that data.
Usable? No idea. I've never tried. So I left that note here so someone trying to make it work knows the data is gone for sure if they strip the letterbox off.