Why do I have black bars top and bottom after exporting my video in FCP?

In the last image you are showing the video in full screen.

What is the aspect ratio of your display?

Open your original movie in Quicktime Player, NOT in fullscreen, make sure that cursor is on the video so we can see the dots on the top left of the window. Post a screenshot - something like this:

I am guessing that maybe there are no such black bars, it is just that the video does not completely fill your display in full screen, due to having different aspect ratio.

The black bars NEVER existed... they did not exist in the original, nor the exported movie.

This is a simple matter of geometry. Not all rectangles are similar. Some are wider, some are taller.

If your video does not have the same proportions as the screen, there will be black areas on the sides, or at the top and bottom, depending.

To illustrate, suppose you made a square video, but your screen is not square - it is (usually) wider than it is tall. So if you try to fill the screen, there will be black bars on the sides:

On the other hand, the same video on a portrait screen (taller than it is wide) will look like this:

My "video" is all green, it has no black bars at all, it just does not match the proportions of the screen exactly.

The same happens with yours.

Your monitor isn't the same aspect ratio as the 4K file. When you view in full screen the image is displayed from edge to edge, so nothing is cut off on the sides, the top and bottom of the screen is filled with black.

You might want to make a custom size project that would allow you to create an image that eliminates the black pillar bars in the exported file, though you will see it in QuickTime full screen, but no black at the top and bottom.

Can you give the frame resolution of the media and of the project properties?

In FCP Settings change the view in Playback settings to Checkerboard.

The last image is what you see when you export and open the file with QuickLook?

Not full screen in QuickTime. What you see depends on your monitor. Use QuickLook. Select the file in the Finder and press the spacebar.

You have a mix of 16:9 and 4:3 footage in your project? The 4:3 screen shots of the gatherings came from VHS?