The permittivity of the vacuum is just to accommodate a pretty unnatural choice of units for electromagnetic phenomena, the SI units, to a natural choice of units like the Heaviside-Lorentz units. The only universal physical constant entering electromagnetism as compared to Newtonian mechanics is the socalled "speed of light", which should rather be called the "universal limiting speed" of relativistic spacetime.
https://en.wikipedia.org/wiki/International_System_of_Units
The most natural choice of units in relativity is, of course, to measure time intervals in the same units as space intervals, using the universal limiting speed. Indeed that's realized in the SI since 1983, because in the SI one defines the unit of time by a certain hyperfinestructure transition in Cesium (since 1967):
The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
Then the metre as the basic unit for length is defined by fixing the value of the limiting speed:
The distance traveled by light in vacuum in 1/299792458 second.
This strange number, of course, only comes about to keep the metre at the same length as its original definition as ##10^{-7}## of the length of the meridian going through Paris. Nature doesn't care much about this distance, and thus it's not a natural unit and not reproducible easily (the story behind defining the metre by measuring the length of the meridian is however a very exciting adventure story, quite dangerous for the brave scientists doing right this).
The permittivity of the vacuum, ##\epsilon_0##, in the SI units for electromagnetism is totally artificial. While to keep the speed of light ##c## in usual units like m/s makes some sense, because we are used to measure times in seconds and distances in metres, the introduction of a fourth unit in electromagnetism (in the SI the Ampere for electric currents) is totally artificial, because it makes us measure the electric components of the electromagnetic field in a different unit than the magnetic components, but relativity tells us that there is only one electromagnetic field, described by an antisymmetric 2nd-rank Minkowski tensor, the Faraday tensor ##F_{\mu \nu}##. This is analogous to the usual system of units used in the US, giving distances in miles and height's in feet and inches. Of course the SI units are just chosen to have convenient numbers when dealing with everyday applications in electrical engineering and not for fundamental theory of electromagnetism. In theory thus the Gaussian system of units or sometimes its rationalized version, the Heaviside-Lorentz units (the latter used particularly in high-energy particle theory since in these units QED uses the same conventions as the more modern siblings QCD and QFD, making together the Standard Model of elementary particles).
The same concerning the units is true for quantum theory. Then another fundamental constant enters the game compared to classical physics (classical usually are called all physical theories not taking quantum theory into account, no matter whether you use a Galilei-Newton or a relativistic space-time model) is Planck's unit of action ##h## or more modern the modified Planck constant ##\hbar=h/(2 \pi)##. In a natural system of units you set ##\hbar=1##. Together with ##c=1## you've only one unit left, for which in high-energy particle theory one usually chooses GeV (giga electron volts) for masses, energies and momenta. In this system length and time intervals are measured in 1/GeV. You can as well measure lengths and times in some unit (usually fm, i.e., ##10^{-15} \;\mathrm{m}##). To switch back and forth between these choices, you only have to remember that in usual units ##\hbar c \simeq 0.197 \; \text{MeV} \, \text{fm}##.
To get finally rid of this remaining arbitrariness in the choice of units, you can take also general relativity and gravitation into account. There the fundamental constant is Newton's Gravitational constant ##G##, which you may choose to set to 1. Then all quantities are given as pure numbers. However, this (modified) Planck system of units is not in common use in physics (except sometimes in the General Relativity community), because the numbers become pretty inconvenient since the Planck length is so tiny and the Planck energy scale so huge:
https://en.wikipedia.org/wiki/Planck_units