How do I install a cabinet door underneath a breakfast bar?
I think I get the picture without one :) I've seen this many times. I'm assuming the wall under the bar you want to cut into is a typical wood framed, wallboard sheathed stub wall. If not, ignore this answer.

You first need to cut out the outer layer of wallboard. If done carefully, no additional finishing will be required, though it may be easier to not be careful and just refinish. It would depend on how easily you think you can match the existing finish. Things will be easier if you plan the opening to fit between two studs, but the opening will possibly be rather small and possibly not in the ideal location. To do things for an ideal size and location, you will likely need to cut out a stud and frame in horizontal 2x4 plates at the top and bottom.

If you're refinishing, remove wallboard from floor to bar top underside and frame the opening properly so the horizontal plates support the new wallboard edges. If you do not want to refinish, place the plates below and above the wall board edges, leaving the wallboard edges unsupported and the plates exposed, they will be mostly covered in the end.

Cut out the wallboard of the opposite surface that is the back wall of the corner cabinet. This time all the edges can be supported. Now you have full access to the depths of that cabinet, but no door. The door can be anything, what I've usually seen is a cabinet face frame applied to the surface of the wallboard, which will cover the exposed horizontal plates. Then a standard cabinet door is attached to the face frame.

If you have decent woodworking skills, you could build the face frame and order the door from a cabinet supply. If not, you could order a plain face frame to match the cabinet door.

It's also possible to case the opening and install a hollow core door, sort of like a little closet. You'd need wide casing to cover the exposed plates. A simple panel door made of MDF could be fabricated and painted. You could just hang some fabric drapes over the opening. The possibilities are only limited by your creativity!