Dave Chase must've been aboard a naval vessel! His explanation is the best so far, but he's kinda got it upside down.
For the most part, you walk on the steel that seperates the decks, or rather, whatever kind of floor covering they put on it, which may be as little as paint or non-skid, or as much as tile, decking, or even shag carpetting. True, there are places where you have things under the deck, but 9 times out of 10, you will have had to step up onto it. This step is usually about the height of a single step, or about 9 inches (~20 cm).
The air conditioning ducts, water pipes (hot water, cold water, and drinkable and non-drinkable vesions of these), steam pipes, fuel pipes (perhaps multiple kinds of fuel, depending on what kinds of subordinate craft you carry), oil lines, telephone lines, network lines, bracing for each frame (a frame is a section of the ship which is typically about 10-20 feet along its length, but there is no hard and fast rule), firemain(s), and if the crew is industrious, there may be lines for their own personal networks so they can play Unreal Tournament during Med Cruise.
Obviously, an aviation ship will have more fuel pipes running around and probably more fire suppression lines. Surface combat ships in the nuclear age will have a sprinkler system for washing contamination off the hull. Amphibious and submersible ships will have extra piping for pumping ballast water in and out.
Some of these pipes are 18" or more in diameter!
I would imagine that in the future, pressure-sensing equipment would be cheaper and more sophisticated, and would be incorporated into an iris valve's installation if that valve was in a major bulkhead. Such a gauge would tell you if there was a huge pressure differential or not. Modern warships don't do this because the pressure differential between their "clean" sections and "dirty" sections is not so great, maybe a quarter of an atmosphere. You just need to have a firm grip on the door to keep it from smacking you, if you're in too much of a hurry to follow the proper airlock proceedures.
Most if not all passageways will have a headway clearance of 7 feet, which is more than enough for most people. Ships designed for taller or shorter people will be correspondingly different.
That upper 3 feet or so seems big on paper, but in practice, when you have to get up there and clean it, it can be quite crowded at times and quite open at others. Either way, you need a stepladder to get up there, and if you're sliding yourself over a big vent to get at another one, you may be standing on the top of that shaky 6' monster.
