Without getting too technical, you cannot create a solid object as you describe in Blender. 3D is already a lie for the most part (basically points and surfaces in virtual space projected to give depth etc). OpenGL, the software interface to graphics hardware that Blender uses draws primitives — points, line segments, polygons etc. and primitives are defined by a group of one or more vertices. In most 3D applications, models are represented as shells or cases where only the surface matter, this makes it easy to edit them for one.
However, there are several ways to fake this in Blender such as using booleans or the bisect tool. These work in a way such that when a part of the mesh is subtracted, the surface(s) of the affected mesh(es) is closed and this gives an illusion of an object being truly solid.
Using booleans (dynamic if you use the modifier)
Using the bisect tool (takes multiple objects into account)
** Images from the Blender wiki.