High enough to get the roost noticeably higher than anything else, especially higher than the nesting box. High enough so you can have a vent over their heads when they are roosting and them not be in a draft. I don't know your style, configuration or layout so I cannot give you specific numbers.
The nest box needs to be at least 12" x 12". I needs to be tall enough that you can put a lip on the bottom about 4" high to keep them from scratching the nesting material and eggs out of the nest and still have an opening about 7 or 8" high over that for them to get in. It depends how you frame it up as to how high it actually needs to be.
I don't know where OC is so I have no idea what your weather is like. If it is extremely cold, a coop on the ground is usually warmer. If your climate will allow it, a raised coop the size you are talking about is usually a lot easier to clean and manage. Even if you made a 4'x4' tall enough to walk in, you would not have enough room to do anything.
OEG usually do not handle confinement well so they will need more room in the run than many other breeds.