Impenetrability is defined as the property that no two objects can occupy the same place at the same time, a definition which can be attested by innumerable sources. If a class of objects have this property, they then are impenetrable. This is the traditional definition of Impenetrability, which has been considered a property of matter since the time of Plato. Now, given the apparatus of modern physics I will defend the proposition that the definition before given is an adequate conception of Impenetrability according to modern physics, as long as objects are properly defined. I will defend it first from the classical paradigm (of both Newtonian Mechanics and the two Relativity theories), and then I will move on to the vastly more intriguing and complex case of quantum mechanics.
For classical theories, it is really quite simple. No matter whether you discuss Newtonian mechanics or General Relativity, it is taken to be axiomatic that a particular point in space-time can only be occupied by one 'packet,' or infinitesimal of matter. This means that the set of positions in space time occupied by a piece of matter cannot overlap with the set of positions of a different piece of matter. Hence, not only does the traditional definition of impenetrability make sense in classical theories, matter obeys the principle of impenetrability according to which ever theory you chose. However, there is one complication. Since different parts of an object move at different speeds, the set of positions of a piece of matter at a particular time is hard to define, since we have to decide which reference frame we are using. However, whichever reference frame we use, the property of impenetrability will hold; it is a generally covariant principle.
For Quantum Mechanics, things get vastly more interesting. Impenetrability is not in the least bit axiomatic in Quantum Mechanics and in Quantum Field Theory, and needs careful examination as well as the application of certain principles as well as a definition of matter for impenetrability to hold. This can be considered a consequence of the strangeness of Quantum Mechanics. Now in Quantum Mechanics, there is a principle called the Pauli Exclusion Principle, which you have stated; while this principle is necessary for establishing impenetrability it isn't sufficient (the uncertainty principle is also necessary). The technical formulation of this principle is as follows: the total wave function for two identical fermions is antisymmetric with respect to exchange of the particles. Before we decode this formulation, it is important to note that there are certain bosons with mass such as the W and Z bosons, which by virtue of obeying Bose-Einstein statistics violate the Pauli exclusion principle. These particles, which mediate the weak force, do not in fact display the properties of impenetrability; hence objects with mass (matter) do not necessarily have the property impenetrability. Back to the Pauli exclusion principle, its technical formulation implies the following informal one: no two identical fermions can occupy the same quantum state. This property has been used in conjunction with the uncertainty principle to show that ordinary bulk matter is solid and occupies volume; solidity of course implies impenetrability as the British philosopher John Locke observed. This idea was first suggested by the great Quantum physicist Paul Ehrenfest, here is a nice explanation why from Lubos Motl from physics stack exchange.
In summary, the property of Impenetrability as defined traditionally is adequate, with the additional caveat that according to the non-classical Quantum Mechanics this property belongs solely to ordinary matter and to a lesser extent to fermionic matter; there are exotic matter particles such as the W and Z bosons which don't display the property of Impenetrability.