Do electrons have some volume, area or shape?
Quoting from the Nobel lecture of Hans G. Dehmelt (1989):


  With the rise of Dirac’s
  theory of the electron in the late twenties their size shrunk to mathematically
  zero. Everybody “knew” then that electron and proton were indivisible
  Dirac point particles with radius R = 0 and gyromagnetic ratio g = 2.00. The
  first hint of cuttability or at least compositeness of the proton came from
  Stern’s 1933 measurement of proton magnetism in a Stern-Gerlach molecular
  beam apparatus. However this was not realized at the time. He found for
  its normalized dimensionless gyromagnetic ratio not g = 2  


...  


  Similar later work at still higher energies found 3 quarks
  inside the “indivisible” proton. Today everybody “knows” the electron is an
  indivisible atomon, a Dirac point particle with radius R = 0 and g = 2.00....
  But is it? Like the proton, it could be a composite object. History may well
  repeat itself.


So there is no evidence that the electron is anything but a point particle, yet a non-zero size has not been ruled out either.  

New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron Phys. Rev. Lett. 97, 030801 (2006) (more-official link) shows that any radius of the electron is less than $10^{-18}$ meters.  

See also the related webpage Limit on Electron Substructure