The "radius" of an elementary particle is not as simple a concept as the radius of a baseball. An electron is described by a wavefunction, which in principle covers all space. There is a way to define a size anyway, but you can imagine that it is not a clear-cut matter.
Also, particles are called elementary when their interactions show no internal structure. To an extent, is it as asking if the wavefunction has only one "central bump" or more.
What ahrkon states is quite correct. Since all the information we might extract is encoded in the wavefunction we cannot ascribe a radius to the electron. The wavefunction is always spread out over some volume. In a metal for instance the wavefunction of an electron is a plane wave that spreads out over the entire volume of the metal. Strictly speaking, the electron is just as large as the metal you're looking at.
The lengthscales in such a problem are defined by the wavelenghts of the particles involved. In the metal the wavenumbers of the electron states are more or less given by k=n*pi*a/L, n=0,1,2,.. a is the lattice constant and L the total size of the system. The smallest lengthscale is thus the lattice constant. In a superconductor the electrons form Cooper-pairs. These cooper-pairs are far more spread out (even though they are composite objects they are regarded as a single (quasi) particle) with a "coherence length" of up to a couple of microns.
In the condensed state it is often not even useful to speak of "a" electron. In solids there are no longer electrons, but "dressed" objects called quasi-particles. An example of such an object is the Cooper-pair, or the small polaron.
In any case, it is not really useful to speak of the size of an elementary particle. This is a little different in string theory (not a subject i'm really familiar with) where particles are represented by one dimensional objects (loops). If i remember correctly these can have a size, but given the fact that these things live at the planck scale they are probably a lot smaller then 10^-15 m
Heumpje refers to the Copenhagen Interpretation of Quantum Mechanics and its understanding of a particle ("particle is a wave").
In the web site mentioned by Adam another approach was taken, following the original particle concept of Louis de Broglie:
One can use the magnetic moment of the electron and its deBrogle frequency f following from its mass/energy (E=h*f), to calculate the size of it. The resulting size is as given there, it is much greater than the conventional assumption, true! You can then use this size to calculate the mass of an electron, using the same equations. This calculation yields the correct value for the mass of the electron within a margin of 10^-3.
This quite precise result which is not available by the current physical main stream shows, that this approach cannot be just speculation.
The conclusion from the past experiments that an electron does not have an internal structure is not correct. If the electron is structured as assumed in that theory the present experimental methods are not able to identify its internal structure.