Technically, they can be emulated but with an exponential slowdown.
An representation of n qubits for example demands that you store the complex probability amplitude of each of the 2^n possible classical states (this is the "state of the n-qubit quantum system"). There is a differential equation involving a 2^n * 2^n matrix that you have to integrate over time to find that state at time T. Finally you have to make the solution classical (this is easily done though, it consists in projecting the 2^n vector onto various axes to get the classical probabilities, then select a classical solution according to these probabilities). Overall intense on the CPU and memory though.
For infinite-dimensional states (e.g. position of a particle in space where each position is one dimension) things are even nastier.
All in all, a good reason to good back to the good old principles of analog computation.
If you think about it, classical computers are basically just NAND gates glued together, it's not surprising that they fundamentally can't do everything...
Please read up on the Theory of Computation, especially the Church-Turing thesis and on Complexity Theory, especially the classes P and NP and BQP. The latter can be gainfully learned about here.