The amount of energy required to break the Coulomb Barrier This web page says that only 3-10 keV of energy is required to break the Coulomb barrier for colliding hydrogen nuclei.
Pretty far from what originally said.
Is this true? Does this have something to do at all with cold fusion?
P.S: I'm just a Physics enthusiast, very lay.
 A: While the figure seems relatively low (and it may be a mistake by an order of magnitude), it is still way above $O(1\,eV)$ which are the chemical energies of electrons in atoms, so can't occur as a thermal fluctuation. It's almost excluded that any single atom at reasonable temperatures would collect enough energy to get through the barrier (although tunneling may help a little bit so that one doesn't have to go "quite" through it).
However, the figure 3-10 keV is a big underestimate for realistic situations. Note that two protons can't merge because there is no Helium-2 isotope ;-) (a diproton would actually exist if the strong nuclear force were just 2 percent stronger). You need at least one heavy hydrogen nucleus to get the fusion going and the barriers are always higher which is why the figure 3-10 keV is unphysical, whatever it exactly meant. The Coulomb barrier goes up as $Z_1 Z_2$ for larger nuclei (in fact, even a bit higher than that). Already for the commercially promising deuterium-tritium pair (which are still just "heavy" hydrogen nuclei), the barrier is about 0.1 MeV. For any larger nuclei, the barrier is already of order 1 MeV or several MeVs, comparable to the energy gain from a single nucleus.
The energy of inner shell electrons can be as large as keV's but that would only be enough for the (non-existent) light hydrogen fusion; however, the inner shell electrons only exist around larger nuclei where the required Coulomb barrier is higher. It can be seen that the Coulomb barrier between the nuclei is always larger than the energy carried by the electrons, even by the inner shell electrons. The Coulomb barrier between the nuclei is really in MeVs or close to it while you can't get above keVs for inner shell electrons.
Some concentration of energy is needed for fusion. When an MeV energy is concentrated into one particle, whether it's a nucleus or a "catalyzer" in any hypothetical design, it's legitimate to calculate that this particle has the temperature of many million degrees Kelvin according to $E\sim kT$ so it can't be "cold" fusion. At least some particles near the hypothetical fusion have these huge temperatures and when they help the nuclei to merge, the relevant nuclei will have this huge temperature as well. So not only a few particles but a "whole locus" of the nuclear material is inevitably very hot. Of course, one may try to make the "hot place" very small geometrically (e.g. by focusing many lasers to one point) but there has to be a hot place for the fusion to proceed.
A: This is correct--- you need a few KeV to break the Coulomb barrier and allow fusion. But a few KeV is still well above the energy of ordinary chemical reactions, and well above the energy of thermal motion, so at ordinary temperatures, when the typical thermal energy of a molecule is about 1/30 eV, 3KeV is 100,000 times bigger, it's just enormous. There is no probability of getting KeV's in a single particle through thermal motion.
Still, 3KeV is the energy of inner shell electrons, and inner shell dynamics can conceivably be responsible for cold-fusion--- the no-go arguments are all circumvented when the inner shells are excited. This shows that the standard theoretical arguments against cold fusion are no good, but it doesn't explain cold fusion by itself.
There are a handful of beam experiments, where people shine a beam of deuterons at energies of 1-20 KeV into a deuterated metal, and look for signatures of fusion. These experiments are notable, because there are unexplained enhancements in the fusion rate at low beam energies, by a factor of about 3 or so (not so big by cold fusion standards). The actual beam measured cross section for hot-fusion is still too small to be responsible for cold fusion, but cold fusion can't be hot fusion anyway, or Pons and Fleischmann would have died from the neutron flux.
In cold fusion, the major mystery is how you output alpha particles without neutron emission. Since this process violates momentum, it requires a spectator body, a spectator electron or a spectator nucleus, to picks up the energy momentum electrostatically. But a spectator electron has a small charge, and a spectator nucleus requires the fusion intemediates to be close to the nucleus. But order of magnitude estimates don't rule out the thing immediately, and some idea must work, because the experimental data is so overwhelming.
A: Yes, this is approximately true and this is why the physics community is so skeptical about the cold fusion.  There is just no way that the electrochemical or crystal lattice pressure could overcome this Coulomb barrier and permit cold fusion to occur.  Extremely high temperatures and pressures are required for any significant amount of fusion of hydrogen or deuterium nuclei.
Extraordinary claims require extraordinary evidence... and they are not there yet.
