I am in the design phase of a chest of drawers project, and it occurred to me that nearly every piece of wooden furniture I have seen has a 2-4 inch gap under it. Often this has the furniture up on short legs, but even where the front/sides come to the floor, the back is open. I am trying to figure out if there is a particular reason for this feature?
The main reason is that woodworking of old days, Joinery was more of an art than a mechanised given. The structure was designed to accommodate for wood settling and fluctuating due to weather conditions (humidity etc) ...
Basically, the most stable structure was one, where the weight rested on the four points which were the most stable (the legs or balls) As Rozwel noted, a flat board on the floor would be susceptible to high points which would unsettle the joinery of the piece as the wood base would most certainly flex to accommodate for the high point and in doing so, throw the piece out of square.
Short answer, the four contact points would lead to a more stable cabinet over a more diverse set of floor types.
It is possible to design a cabinet or chest of drawers without legs at all. However, in traditional cabinetry, the legs are actually part of the structure.
If you study traditional furniture, you will see that these legs have a specific purpose. They typically go from the floor to the top of the piece in each of the 4 corners. The legs are load bearing, and they are also incredibly strong. The cabinet box is then constructed by having 4 horizontal members on the top and bottom which connects the legs together. This form of design goes back many centuries, and has not changed very much.
The rest of the elements of the cabinet should not be load bearing such as the sides and the back. If the legs ended abruptly at the bottom rail and did not extend past it, then non load bearing components would be in contact with the floor. This would both weaken the cabinet and interfere with its operation.
In other words, if you chest of drawers had no feet, then the support for the bottom drawer would be directly on the floor. Since it is not meant to be load bearing, the support will flex and bind the drawer which will make it difficult to open and close. Even if this was not the case, when the drawer is opened, then the bottom of it will be resting directly on the floor. That is something that is not desirable.
The longer legs allows it to evenly distribute the weight of the piece and its contents. Other design elements then would not have to then bear the brunt of the weight.
Having legs also has the added benefits of being able to set it on a floor that isn't perfectly level without rocking or being unstable. Going back to ancient times before wall-to-wall carpeting, it also allows you to sweep under them. Floors in these homes were also fairly damp and the piece would become waterlogged, moldy, and eventually rot.
The piece on the bottom that you mentioned in the question is sometimes called an apron. It is there for aesthetic purposes only. If you look closely, you will see that there is a small gap between the apron and the floor and the actual weight is actually supported by the legs. Since this is purely aesthetic, there is no need to install one behind the cabinet where nobody can see it.
Blindly following what went before. Because people buying furniture don't think a box right on the floor is "furniture". Ratchet Freak is quite right as well...
I suppose at some point in the past it kept them off the mud and dank of the dirt or stone floor, and thus had a functional purpose, largely long-lost.
I rather like the Shaker solution of putting beds on wheels so you can sweep under them, but that does not tend to be the "Shaker furniture" people want to buy. I also like their extensive built-ins (so you'd just have drawers in a wall, rather than a chest of drawers.)
Also provides great mouse habitat, so it may be that we were programmed to like furniture that way by hyperintelligent pan-dimensional beings who's protrusion into our dimension looks like white mice (R.I.P., D.A.) ;-) It certainly serves as a place where cat toys can be stored out of reach (with their dust-bunneh friends) until you get the humans to fish them back out for you.