How do I select an area with an exact width and height from the center of an image in Photoshop?

Turn on the rulers. In my version there is a selection under the View menu.
Make sure guides are visible under View->Show
Drag from the vertical ruler on to the picture to create a guide. As you move it you can see the pixel position. Position guides at the points you want to select. You may have to zoom in a lot to get the exact pixel position.
The rectangular selection tool will snap to the guides.

Another potential way to do this is by changing the canvas size. (I don't have Photoshop, but I believe that this will work - it works in Affinity Photo :D)

1. Select Image=>Canvas Size (which opens the Canvas size dialog box)
2. In the "New Size" section, set the anchor to be the center box
3. Leave the "Height" as is
4. Set the "Width" to be 720 pixels
5. Leave "Relative" unchecked.
6. Click on "Ok"

This should take the 720 pixels around the center of the image and effectively crop out the left and right 600 pixels.

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Down voter care to explain? Does this method not work?

Choose "fixed size" in the Marquee selection tool ribbon and set it to 720x720 (or 720xH for whatever your height is). Then place a selection on the canvas. Then drag it left or right until it centers (the center vertical line should show when it does).