Is space "real", or is it some sort of accepted postulate? What is space?
It seems to be everywhere in the equations of physics, as some sort of postulate or hidden hypothesis. We also have a direct experience of it, but is it "real"? The fact that we experience it doesn't mean that it exists by itself. For example, what we are experiencing as heat or cold is just molecules that are vibrating at different speeds. Hence our sense of temperature is just an emerging effect that can be explained in base of more fundamental phenomena, to make my point.
Is it the same for space? Can physics explain space in a more fundamental way, or is it some sort of accepted postulate? And if physics can explain it, then what is space?
On the same line, what is a vacuum (absence of matter)? Is there a vacuum everywhere in space? And what does it mean or imply? Or, is it that space is dragged in by matter? Or does space manifests itself only in presence of matter? In the latter case, what is the relationship between space and matter that shows that?
Voila!
Apologies for the naive questions, but every time I think about space I'm stuck. I don't get it. Yes, I experience it as everybody else does, but I don't understand how it fits in the big picture, it's almost like we just have to accept it as it is. Maybe it's the case, but I guess you probably have a better insight than me on the subject.
Also, I thought this could add water to the mill. It's a link to one of Sean Carroll's blog posts on the question of whether or not space is fundamental.
 A: I believe the true answer to your question is that our observation of space fits a mathematical model for a 3-dimensional geometry.  My wording is backward compared to historical development, which is why it's hard for people to decouple these things.  Humans learn about the degrees of freedom in the space around them as an infant, and possibly even sooner.  The fact remains, however, that concepts like $\sqrt{2}$ have a consistent formalization within the field of real numbers and a corresponding geometric interpretation (like the diagonal of a square).  There is room for philosophy about what that interpretation means, but it is not an impediment to physics either way.
We've had several occasions to propose revisions to our concept of space and time.  General relativity's conclusions are accepted beyond all reasonable doubt.  Our mathematical models expanded in correspondence with our beliefs about the universe.  String theory also offers revisions that involve much more difficult math, but these are all knowable and calculable.  This doesn't change anything about the basic fact that we fit a mathematical system to our universe.
The only useful thought experiment I can imagine within this philosophy is if the mathematics of geometry would be discovered by mathematicians living in a world without our type of geometry.  That begs the question of whether we have discovered some other mathematical system that could function as an alternative to geometry as we know it.
We have formal systems to work with here.  Humans have developed many formal systems.  With computers, we have been able to study the nature of a wide variety of formal systems due to the low cost of running experiments.  All of what we do with computers fits into a certain class of mathematical system, with is Turing computing.  Within these systems we are able to model real physical systems, which is reasonable, because as I said, geometry itself is a mathematical system, and other physical laws are added onto it as needed.
Turing computing, however, still can't emulate geometry in finite time.  This is a deep philosophical question.  For classical and simple physics, I can plainly state that the Galilean invariance rules out the most logical approaches to a cellular automaton at the root of space.  You can go further into the details of Lorentz invariance to find even stronger statements, but it's not important.  Even aether theories from the turn of the 18th-19th centuries reflected a logical inroad to pixelization of space.  If an aether theory had been true, then a finite Turing machine could have validly been at the root of space.
Of course, those theories are not true.  Reality has been adamant about preserving the laws of physics in all reference frames, which spells doom for theories predicting that space only approximately fits the mathematical system, and the result of some finite number of constituents.  Our finite computing machines have a number of quirky errors in simulating reality, such as energy drift.  In the real universe, truly conserved quantities like energy appear to be perfectly conserved.  This relates to Noether's theorem.  Basically, we observe that:


*

*Conserved quantities are conserved perfectly

*Relativity between reference frames holes perfectly


These two are two sides to the same coin.  So when we observe that space and time are perfectly consistent in our universe, we can say that it's related to the fact that net energy and net momentum are perfectly consistent scalar and vector quantities.
In short, physics is about writing down the rules of a mathematical system that fits the universe.  You can say that geometry (whichever geometry this is) can also be defined with a set of rules.  Following those rules is impossibly difficult with finite formal machines.  So space could be "false", meaning that it's really just an apparent system that exists on top of a more fundamental system underneath, but that more fundamental system would not be any more computable.
A: Albert Einstein is often credited (erroneously, it seems) with saying

The only reason for time is so that everything doesn't happen at once.

and John Wheeler added

Space is what prevents everything from happening to me!

Now, those quotes may sound silly and self-referential, but they are meant to draw you attention to something very, very basic. Things do happen at different times and different places, which means that "time" and "space" are certainly referents for something. 
Your example in which you claim that heat and cold are not really real is silly: they are referents for the subjective sensation associated with exposing your nerves to conditions of high or low thermal motion.
As for your concern that space is "as some sort of postulate or hidden hypothesis", that has manifestly not been true for just shy of one hundred years. Space as a thing may have been assumed unitl the early twentieth century, but at a the very latest Einstein made space and it's properties an explicit subject in general relativity. 
We have measured the warping of space itself in several contexts including micro-lensing and frame-dragging. Efforts to directly observe gravitational wave are underway.
A: There is a difference between effective or emergent degrees of freedom (or fields), and fundamental degrees of freedom (or fields).
For instance, in string theory, "space" and "time" are in fact fields of more fundamental degrees of freedom, that is :
$$X^\mu = X^\mu(\sigma, \tau)$$
where $\sigma$ is a space-like parameter, and $\tau$ a time-like parameter.
Here $\sigma$ and $\tau$ are fundamental degrees of freedom, while $X^\mu$ appears as effective degree of freedom.
Constraint about some symmetries (conformal symmetry) have consequences about fields depending of $X^\mu$. For instance, at low energy, the gravitational field $G_{\mu\nu}(X^\alpha)$ has to obey a Einstein-like equation. So  $X^\mu$ and $G_{\mu\nu}(X^\alpha)$ are effective degrees of freedom and effective fields.
The vacuum may have different aspects.
The quantum vacuum has fluctuations, even in the absence of matter, and it could have non-zero energy (positive for the bosons, and negative for the fermions)
In general relativity, you could have curved space-time without matter.
And, with supersymmetry, you have not only the usual ("bosonic", communtative) space-time, but you have also "quantum" (or "fermionic", or anti-commutative)  dimensions. 
A: Vacuum is, as you say, the absence of matter. What does that mean in a tangible sense? It means that there is no pressure to restrict the flow of matter into the 'vacuum space'.
Space, as in the region outside the Earth's atmosphere, might be composed of different things depending on where in 'space' you are. In the vicinity of the Sun, there could be millions of atoms in any given cubic milimeter of space. Between the galaxies, you would be hard pressed to capture even a single hydrogen atom in a cubic meter.
