How to measure vacuum permittivity? In this question, the first answer (though I don't completely understand that answer) states that $\epsilon_0$ is the proportionality constant in Gauss' law. If that's the case why isn't it assumed to be just "1". This actually leads to the question, how was $\mathbf{\epsilon_0}$ measured and determined, which again beings me back to "What is vacuum permitivity?"
P.S: I made a series of questions, here. But as it was too broad, I was told to form separate questions, but I have linked everything there, in the comments, kindly take a look.
 A: 
In this question, the first answer states that $ϵ_0$ is the proportionality constant in Gauss' law. If that's the case why isn't it assumed to be just “$1$“. 

The constant $\epsilon_0$ can indeed be assumed to be just $1$. In fact, there is a system of units called Heaviside-Lorentz units (HL units) which does exactly that. 
Gauss' microscopic law is 
\begin{array}{ll}
\nabla \cdot \vec E &= \rho/\epsilon_0 &\quad\text{in SI units} \\
\nabla \cdot \vec E &= 4 \pi \rho &\quad\text{in Gaussian units} \\
\nabla \cdot \vec E &= \rho &\quad\text{in HL units} \\
\end{array}
Similarly, Coulomb's law is
\begin{array}{ll}
\vec F &= \frac{1}{4\pi \epsilon_0}\frac{Q_1 Q_2}{r^2}\hat r &\quad\text{in SI units}\\[1em]
\vec F &= \frac{Q_1 Q_2}{r^2}\hat r &\quad\text{in Gaussian units}\\[1em]
\vec F &= \frac{1}{4\pi }\frac{Q_1 Q_2}{r^2}\hat r &\quad\text{in HL units}\\
\end{array}
So the form of the equations of electromagnetism and the presence or absence and value of $\epsilon_0$ is all tied to your choices that you make for your system of units. As you suggest, you can indeed assume that $\epsilon_0=1$ and then you wind up with units like HL units.
This is often a challenging concept for students who are generally only exposed to SI units. Whenever you see a dimensionful constant which appears to be a universal constant telling you about some universal property of nature, typically you will find that constant is actually related to your system of units. There are systems of units such as Geometrized Units and Planck Units that are designed to avoid all such constants entirely.

This actually leads to the question, how was it measured and determined

This is measured by actually measuring the values in Coulomb's law. For example, you can get two objects with equal and opposite charge by using opposite plates of a charged capacitor. You can measure the charge in coulombs on each by measuring the current in amps and the duration in seconds as you charge them. Then you measure the force between them in newtons and the distance between them in meters. Then $\epsilon_0 = \frac{1}{4\pi |F|}\frac{Q^2}{r^2}$
The key to this is to have an independent method for measuring the charge. In other unit systems there is no independent method for measuring the charge. For example, in Gaussian units the same experiment gives you a measurement for the amount of charge as $Q^2=|F| r^2$ and this measurement of the charge can be used to calibrate your current measurement device.
A: As the comment by G. Smith says, you can actually set the proportionality constant to one. But then you would have to measure electric charge in some other units.
Consider the setup of SI units. One coulomb is the charge that is carried by a current of 1 Ampere in one second. An Ampere is defined as the current that causes two infinitely long and thin wires at 1 meter from each other to attract with a force of $2 \cdot 10^{-7}$ Newtons per each meter of the length of the wires. So, this definition is kind of tied to the Lorentz force. When you ask a question like "What is the Coulomb force between two static charges in vacuum?", you get a strange constant.
In the Gaussian units, for example, the situation is different. Here the charge in such a way that the constant in Coulomb's law is equal to one.
In short, if you define the charge so that it "makes sense" in terms of meters, kilograms, and Newtons, you will get odd-looking constants in electromagnetic laws. But if you define the charge units so that electromagnetic laws look nice, then one unit of charge in this system will have an odd-looking proportionality constant to the Coulombs (1 CGS charge unit $ \approx 3.33564×10^{−10}$ C).
A: Please do not accept my answer, but rather the one of Алексей Уваров
I just want to make his answer clearer.
Алексей Уваров'asnwer is really the correct one ! 
The value of  $\epsilon_0$ is really linked to the definition of the Ampere, the unit of current intensity. You might ask, why such a ridiculous number as $2\ 10^{-7}$ Newtons per meter ? Well, the factor $10^{-7}$ is there to make the Ampere a manageable unit. And the factor 2, well, there is a very good reason, but it is a bit hard to explain what it is. Very roughly, because the area of a sphere or radius one meter is $4\pi \ m^2$ while the area of the side of a cylinder of radius one meter and height one meter (not counting the areas of the circles on top and bottom, just the “side”) is $2\pi \ m^2$  and $4\pi/2\pi=2$. No kidding, this is really and truly the reason.
The point is, one has decided that the quantity known as the permeability of the vacuum should be $\mu_0=4\pi\ 10^{-7}$ in the appropriate units. This is, as explained above, a definition of the Ampere. Since the value of  $\mu_0$ depends on the units, fixing arbitrarily its value when all units has been fixed except, till that time, unit of electrical current intensity fixes value of the latter to one Ampere by definition.
Now there is a physical property that can be proven through Maxwell's equations, that the vacuum permittivity $\epsilon_0$ and the vacuum permeability $\mu_0$ are related to the velocity $c$ of light in the vacuum. The relationship is 
$\epsilon_0\mu_0 c^2=1$
So in order to obtain $\epsilon_0$, it is necessary to measure the velocity of light. The permeability $\mu_0$ has been fixed exactly by the definition of the Ampere, it is the value of the Ampere that depends on measurements.
The value of $\epsilon_0$, contrariwise, depends on a measurement. Now it just happens, really by pure chance, that the units of length and time (which were  originally fixed by the french revolutionaries COCORICOOOOOO !! - note that I am french) happened to be such that the velocity of light is almost a round number. 
It is pure chance, it was impossible to measure the velocity of light at any accuracy at that time. It is almost 300000 km/s, but not quite. (Now it has been fixed to exactly 299792458 m/s, by changing the definition of the meter, which is not a fundamental unit anymore, but depends on the unit of time, namely the second, which has now a definition based on some physical property. But they decided to round off the velocity of light to the integer closest to the value previously obtained by using the old definition of the meter, that was previously based on some physical property and thus could not really be measured with perfect accuracy anyway. As you see they did  **not* decide to round it off a 300000000).
Anyway, for most practical purposes, using the very good value 300000 km/s for $c$ one usually uses for $\epsilon_0$ the value
$\epsilon_0 \approx  1/ (36\pi 10^9)$
but note that, not only is it not by definition the way $\mu_0$ is defined, and it is not even the exact value, because the speed of light is not a round number in the SI system.
For some very precise measurements, the exact value of $c$ must be used 
$\epsilon_0 =  1/ (\mu_0 c^2)=1/(4\pi \ 10^{-7} c^2)$
