Can stable nuclei theoretically fission through quantum tunneling? As I understand it, an unstable nucleus is going to randomly fission because the forces binding it together are momentarily weaker than the electrostatic repulsion of the protons.
Given that some nuclei are really unstable, and others are stable for billions of years, is it true that so-called stable nuclei are actually theoretically able to fission through quantum tunnelling, just unlikely to do so?  Have stable atoms ever been observed to spontaneously fission?
Additionally, will bombarding stable nuclei with particles split them?  Can you hit a Carbon atom with a neutron and split it, or will it always add on or miss?  Can a photon split a nucleus? 
 A: Stability of nuclei is a matter of the binding energy and conservation of quantum numbers: number of baryons is conserved. If ( and it is a large if) the proton decays, i.e. baryon number has a small but finite probability of not being conserved, then also stable nuclei might decay spontaneously. At the moment the lifetime of the proton has a limit of larger than 10^32 years.

will bombarding stable nuclei with particles split them?

yes, again depending on the energies involved

Can you hit a Carbon atom with a neutron and split it,

yes depending on the isotope and the energy

Can a photon split a nucleus? 
  Yes, if it has MeV energies, i.e gamma rays.

"The height and shape of the fission barrier are dependent on the particular nucleus being considered. Fission can be induced by exciting the nucleus to an energy equal to or greater than that of the barrier. This can be done by gamma-ray excitation (photofission) or through excitation of the nucleus by the capture of a neutron, proton, or other particle (particle-induced fission). The binding energy of a particular nucleon to a nucleus will depend on—in addition to the factors considered above—the odd–even character of the nucleus. Thus, if a neutron is added to a nucleus having an ... (100 of 9031 words) "
