Quantum theory tells us that all particles also act like waves. It is possible to measure the wavelength of a wave, but the wavelength depends on the energy of the particle.
To see very small things, we use an electron microscope. To see finer details, we increase the voltage of the electron gun, which increases the energy of the electrons, and reduces their wavelength. This changes the "size" and "volume" of the electron.
Waves do spread out (like ripples from a pebble dropped in a pond), and so there is no hard "edge" to an elementary particle - there is just a region where there is a vanishingly small probability of finding the particle.
Particle physicists who collide particles with each other can calculate an effective area for the particles, by knowing the physical area of the beam, the number of particles in the beam, and the probability that particles will collide or pass on by without "shattering". But this size depends on the energy of the particle accelerator.
The size of a Uranium nucleus is approximately 1 "barn":
http://en.wikipedia.org/wiki/Barn_%28unit%29