Save a still frame — Use VLC > Video > Take Snapshot. The directory it's saved in can be found at Tools > Preferences > Video > Video snapshots > Directory.
Cropping black bars off of a video — Handbrake does this easily. Go to Dimensions [tab] > Cropping = "Custom", and tweak the left/top/right/bottom values.
Using the 'Preview' button before 'Start Encode' button can save you a lot of time, and allow you to quickly guess-and-check.
The left/top/right/bottom values aren't absolute pixel positions, rather they're relative to that edge. They're how many pixels you want removed from that edge.
Calculating the size of the black bars — HandBrake tries to automatically determine how thick the top and bottom black bars are (or the left and right bars), but it's sometimes way off. In this case, you have to calculate these dimensions manually:
Open the video in VLC, and use the "save a still frame" steps above to save a static image out.
Open the image in Gimp. Use Image > Duplicate.
To get the width of the first black bar — A) Crop, and leave behind only the first black bar. (We're not going to save this, this is throw-away). B) Image > Canvas size, and record the number listed as either "Width" or "Height", depending. C) Close this image without saving.
To get the width of the second black bar — Repeat the above steps, except for cropping to just the bottom black bar (or the right black bar).