What is Space in Physics? Is space a physical thing like matter? Is it a concrete thing or just an abstract concept?
 A: In the context of general relativity, space is not a physical thing like matter or radiation. However, neither is it entirely abstract. Once you set up an inertial reference frame and a metric, then you can measure distances between locations in space. These distances have physical effects - for example, the distance between two locations determines the minimum time that it takes for information to travel from one location to another, and this minimum time is independent of the choice of reference frame. So space is concrete but immaterial.
A: Your question could be a question about knowledge, how we perceive Nature and what we consider as real things, and more philosophical arguments.
Broadly speaking, in Physics as the science of Nature, following the scientific method by Galileo, as far as we can measure it in a reliable and repeatable way, a thing is a physical entity.
In the limits of classical physics, space is an invariant physical entity (i.e. independent from other physical quantities, and from the observer who describes the phenomenon) and we can use rulers and set squares to measure distances and angles or other more advanced instruments: we can use different instruments but we can agree on the measures and repeat them within the precision of each instruments.
Thus to be precise, we can't measure space itself but distances and relative position/orientation between points and objects in space.
In Einstein's relativity, space  must be considered along with time in space-time to be an absolute/invariant with respect to the observer/coordinates used to describe a physical process, or Nature. When the effects described by Einstein's relativity theory are relevant in a physical process, two observers (even inertial observers, in relative motion) do not agree on measurements of space distance between two events, but they do on measurements of distance between two events in space-time.
So space can be interpreted here as a projection of the physical entity space-time.
A: I think the space is a "concrete thing" because it expands. Its physical measure is volume, I guess.
The expansion of space might be related to creation of the new matter (e.g. particle-antiparticle pairs at the event horizon of the black holes, etc.). Due to expansion of space, the positive energy of the new matter is compensated by the negative potential energy of gravitational attraction of the massive objects.
A: Space is as physical as anything. You measure it with a ruler. The fundamental objects of physics are those you can measure with simple instruments (clocks, balances, force gauges, thermometers, ...). Both the abstractions of theory and the concrete measurements made with more sophisticated instruments are founded on this base.
A: You assume that matter is a simple and straightforward thing to understand, and also that material things, tangibly made out of solid matter, are the most obvious paradigm of "thingness." And I agree with you, because I am also a human being that grew up on this planet, and for me too such are the primary givens of childhood experience... being most intimately connected to everyday life, they seem the easiest to understand. (Moderator: I refer to personal experience here only metonymically, of course I mean everyman not just myself.)
It is not so.
Yes, spacetime is physical, just like matter. The more you learn about it, the more mysterious all of the above will become. The manner in which we get an intellectual handle on any of these physical "things" (or somethings) necessarily involves abstraction, and we are guided by concrete experiment/experience which of course also evolves from our primary senses to highly advanced equipment.
A: Space in general relativity (the physics theory most interested in its properties) is usually treated as an aspect of the "spacetime manifold": a 3+1 dimensional set of points ("events") that locally have relations that makes it possible to speak of neighbourhoods, distances and angles (a "metric topological space").
This manifold is more like a thing than an abstraction in general relativity. Its shape is determined by the field equations, phenomena like matter fields are described as functions of the spacetime points (i.e. "living on" the manifold).
Note that not all physical theories make space into a thing like this. For example, one can formulate Newtonian mechanics and special relativity in terms of mutual distances between objects and treat the intervening space as a mere abstraction. However, field theories generally care about every single point of space, making it the backdrop on which they interact.
A: In physics, space is a concept that refers to the three-dimensional expanse in which all matter exists. It is often considered to be a void or emptiness, but this is not entirely accurate. Space is a physical thing, in the sense that it exists as a framework or structure within which all matter and energy exist and interact.
However, space is not a concrete thing like matter. It is not made up of particles or atoms, and it does not have a definite shape or size. Instead, space is an abstract concept that is used to describe the relationship between objects in the universe and the forces that act upon them.
In other words, space is the framework within which all physical phenomena occur. It is the "stage" upon which the events of the universe play out. While it is not a concrete thing like matter, it is a fundamental aspect of the physical universe and is essential for understanding the behavior of matter and energy.
A: As your question is so wonderfully open to interpretation, I'll interpret it in a philosophical or ontological way.

*

*Except in very advanced parts of physics (anything with "quantum" in its name) space does have properties but is pretty clearly not a "thing" in the sense of an "object". You have the tag aether in your question, and space is definitely not that, by all our current theories. Space is not a bunch of very thin water or air.


*Since space does have properties in many areas of physics, for example lengths, volumes and such - even non-constant in Einsteinian physics - it is certainly a mathematical object. Note that the causation goes from reality towards mathematics, not the other way round. Only because we can describe space using mathematical objects does not mean that space "is" something.


*As of 2022, nobody knows what space actually, "really" is, in the sense that we know a bit about what matter is. For matter, light, and other stuff in the universe, we have a good theory (based on the 4 fundamental forces  or interactions our physics is all about these days, of which electromagnetism accounts for basically everything we can see or experience on a human level). We have nothing comparable for space itself. Yes, relativism tells us a lot about how space behaves when it interacts with matter or energy, but we have no idea what it is in the sense that we know that an atom is something which consists of a nucleus and an electron cloud and so on and so forth.


*It seems intuitive to me (as a layperson) that there must be more to space than it just being total emptiness in which light, matter etc. flutter around, simply because of some of the advanced quantum-based effects like spontaneous creation of particles/antiparticles literally out of nothing (Quantum Fluctuation, possibly proven by the Casimir Effect). Eventually, physicists working on the Grand Unified Theory might figure it out, but even then I could very well imagine that our then-new knowledge about space will be so esoteric that it will be very hard to bring down to the level of everyday people.


*There is a great little video on the question "Why?" by Richard Feynman which gives a good reasoning on why even though theoretical physics might eventually figure out the formulae surrounding Space, we will be no closer to knowing what it actually is (in short: when asking why (or in this case, what) something is, any answer one can give invites more detailed follow-up questions, with no bottom). Even if that interview is rather tangentially related to this topic, it's a great background when thinking about what physical reality actually "is".
