The reason why the numerical value of the speed of light is exactly:
<br />
c = 299,792,458 \, \frac{\mathrm{m}}{\mathrm{s}}<br />
is because the speed of light is used today to define the unit of length through the unit of time and all uncertainty in its determination is attributed to the uncertainty with which the standard of length had been defined. It is chosen this specific value so that it coincides with the old etalon of length in the metric system chosen to be:
"one ten millionth of the length of the Earth's meridian from the North Pole to the Equator passing through Paris..."
and the unit of time (the second) chosen to be:
"1/86,400 part of the mean Solar day"
Today, the second is defined by counting a particular number of periods in the line of a particular isotope:
"the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."
Again, the weird numerical factor is because of the tendency to adhere to the old etalon of time.
These units are terrestrial and have no deep cosmological meaning.
All mechanical quantities have dimensions with respect to length, time and mass. Therefore, there can be no more than 3 freely definable fundamental constants. In principle, one can then choose these constants to be exactly equal to 1 (or another convenient numerical factor, but, nevertheless, dimensionless).
The modern theories developed in the XX century, the Theory of Relativity and Quantum Mechanics naturally establish two such constants: the speed of light in vacuum and the Planck constant. In natural units, the values of these are chosen to be exactly \hbar = c = 1. Because of the dimensions of these constants in SI:
<br />
[c] = \mathrm{L} \, \mathrm{T}^{-1}<br />
<br />
[\hbar] = \mathrm{M} \, \mathrm{L}^{2} \, \mathrm{T}^{-1}<br />
we see that by choosing these constant to be dimensionless, we choose a particular relation between the base physical quantities' dimensions:
<br />
\mathrm{L} = \mathrm{T} = \mathrm{M}^{-1}<br />
(Almost) all electric units have an additional dimension with respect to electric current (in SI). However, because of the way the unit of current is defined - through specifying what the force, a mechanical quantity, between a particularly simple arrangement of conductors is - there is a "fundamental constant", permeability of free space (the magnetic constant of classical vacuum), with an exact value:
<br />
\frac{\mu_{0}}{4 \pi} = 10^{-7} \, \frac{\mathrm{N}}{\mathrm{A}^{2}}<br />
This constant is used to just make the equations of (classical) electrodynamics dimensionally consistent. There is another related quantity, permittivity of free space (the dielectric constant of classical vacuum) \epsilon_{0}. According to Maxwell's equations, the speed of electromagnetic waves in free space is exactly:
<br />
c = \frac{1}{\sqrt{\epsilon_{0} \, \mu_{0}}} \Rightarrow \epsilon_{0} = \frac{1}{\mu_{0} \, c^{2}}<br />
According to the electromagnetic theory of light, light is nothing more but electromagnetic waves with specific frequency (of wavelength). Therefore, this speed must be also the speed of light in vacuum. But, as we just saw, this quantity is a fundamental constant with an exact value. It follows that the permittivity is also an exact numerical factor. In equations of electrostatics, this constant enters in a different combination:
<br />
k_{0} = \frac{1}{4 \pi \epsilon_{0}} = \frac{\mu_{0}}{4 \pi} \, c^{2}, \; [k_{0}] = \mathrm{M} \, \mathrm{L}^{3} \, \mathrm{T}^{-4} \, \mathrm{I}^{-2}<br />
the Coulomb constant.
Another fundamental quantity is the electron charge e ([e] = \mathrm{T} \, \mathrm{I}). We can use the above Coulomb constant to construct a "mechanical quantity" by canceling the dimensions of electric current. Specifically, the combination k_{0} \, e^{2}. The dimensions of this combination are [k_{0} e^{2}] = \mathrm{M} \, \mathrm{L}^{3} \, \mathrm{T}^{-2}, which is a dimensionless quantity in natural units. Actually the dimensionless quantity is the following combination:
<br />
\alpha = \frac{k_{0} \, e^{2}}{\hbar \, c} \approx \frac{1}{137.0}<br />
the fine structure constant.
