Not true, you may have been thrown off by the fact that s orbitals are generally shown as spheres; but the radial extent (probability amplitude) of the wave function goes to zero for small radii, for all s orbitals, see http://www.chem.umass.edu/people/botch/Chem121F06/Chapters/C....
A classically-behaving point particle with the same charge as an electron would drop into the nucleus, but the wave nature of real electrons forbids it. Due to the uncertainty prinicple, there is a nonzero chance of finding the electron in the nucleus, but to say it is substantial is false.
Well, technically, in an s-orbital there is a substantial, non-zero chance of the electron being inside the nucleus.