This question leads us quite quickly to the metaphysical question of what it is that distinguishes physics from pure mathematics. That question is quite subtle so I do not think you will find a complete consensus, among experts, on the answer to your question.
To understand physics generally, it becomes more and more useful, as you progress to more subtle areas, to make clear in your mind the distinction between the mathematical abstractions, such as number, vector, tensor, Hilbert space, quantum amplitude, and the physical things such as bricks, cats, light pulses, planets, electrons, and their physical properties such as mass, charge, velocity, acceleration, and so on. When we write a symbol such as $\bf v$ and say "it is the velocity" then on the one hand we are referring to a mathematical abstraction---a vector that lives in an abstract space of mathematical abstractions---and on the other hand we are referring to a physical quantity: the rate at which an object is increasing its displacement relative to other objects. In Newtonian physics you can get away with being a bit hazy on this distinction, but in quantum physics you cannot. A quantum operator such as
$\hat{\bf p}$ is not "momentum", it is "the momentum operator" or "the operator representing momentum". It is a mathematical abstraction. The physical quantity here is the amount of "oomph" an object may have owing to its mass and motion. That is not an operator but a property of the physical world, one which can be detected by physical apparatus.
In General Relativity there are reasons to say that spacetime is purely an abstraction, placing it at the mathematical end of the distinction, and also reasons to say it is somewhat physical. It is somewhat physical because, after all, it has measurable properties such as curvature, and it can transmit energy and momentum from one place to another. But its physical nature is nevertheless subtle because you cannot tell whether you are moving relative to spacetime, nor even if you are accelerating relative to spacetime---those very phrases do not have any meaning, as it turns out; they amount to a misuse of terminology.
And spacetime is not a thing in the sense of "made of stuff which has energy" because the energy we associate with spacetime (the energy which appears and disappears when gravitational radiation is emitted and absorbed) does not contribute to the stress-energy tensor in the Einstein field equation. This makes the role of spacetime unique among the sorts of phenomena we study in physics.
Another way to argue that spacetime is physical, not merely a mathematical tool, is that you can study many differentiable manifolds using pure maths, but only one of them will be able to give a precise description of physical goings-on as observed in the universe. What singles out this manifold from all the others? If you take the view that spacetime is purely a mathematical tool then it is hard to answer---logically, the special manifold would have to have something further about it that shows that it has this special role, but pure maths is unable to determine that special further property. We can avoid this puzzle by asserting that there is a physical "thing"---i.e. physical spacetime---and the special manifold is the one which correctly describes the nature of this physical "thing".
In view of all this, I for one feel that it is more helpful than not to say that spacetime is indeed a physical "thing", as long as one keeps in mind the various caveats which people who take the other view will also wish to emphasize.
