Yeah, time itself cannot really be measured directly, we can only measure the motion of some physical thing and compare it to the motion of another physical thing.
To reply to the original question, basically what Jesse said; the framework of relativity does imply a static block of 4D-spacetime to be a valid model of reality.
Anyhow, a while back there was a thread that I started about this same issue, and after some confusing exchange, I think the consensus was that there is no real consensus, and that the ontological interpretation of relativity is very much open in this respect.
But I think this is one of the more interesting facets of relativity, so let's talk about it :)
Consider yourself in a lab frame looking at a clock one light-second away. Ask yourself, "what is the actual state of the clock at the moment?" Ordinarily you might be inclined to say the actual state of the clock is 1 second ahead of the state you see (in your "subjective" reality), but you must also assume the clock exists in different state in the "subjective reality" of any observer that is moving in the lab-frame, even if this observer is right next to you.
Basically if you jump onboard of the moving observer now, it could be said that the subjective reality around you should also change its state accordingly. For example, we could just say that the clock went from +1 seconds to +0,5 seconds (half a second backwards) during the change of direction. Of course you wouldn't see any backwards moving clock, but this is the conclusion you'd come to if you assumed there really is a subjective "now"-moment around you the way it is implied in the timespace diagrams, and that framework of relativity is true.
But the math of SR doesn't necessarily say this, or much of anything about the reality of the "present" moment around an observer. It's how Lazycai said; the horizontal line (the subjective "now"-moment) may well be just a product of our imagination, just something that exists in our model of reality.
But I would add that I think Einstein believed there is a subjective "now"-moment. This is implied in the thought experiment about the relativity of simultaneity (with the moving train), and in all the talk about length contraction. (If there is no subjective "now"-moment, then there would be no subjective length-contraction either)
And if you assume there is a subjective "now"-moment (= a 3D-slice of the block-time), then what this would mean to the nature of past and future is that they must be fixed, and anything that happens in your future has already happened in the subjective reality of some observers in relative motion, and also things you've already done may not have happened yet in the subjective reality of another observer.
Elaboration; Let's say you are in a lab frame with a pole at some distance from you. You send another observer towards the pole (at great speed) and clap your hands when you *know* (by figuring out the speeds and distances) the observer is just about to pass the pole.
But when this other observer later figures out when did you clap your hands (from the isotropic speed of light), he will conclude you had not yet clapped your hands even well after he had already passed the pole.
So it can be said that when you had clapped your hands and you knew the observer was at the pole, the reality around that observer was such that you had not clapped your hands yet.
Or other way around, if there is an observer approaching you, and you decide to clap your hand at the moment you figure the observer must have just passed the pole (in lab-frame), then it will turn out that in the subjective reality around the observer you clapped your hand well *before* it had reached the pole. (assuming he won't stop at the pole but continues onwards until he observes your clapping)
So we can say anything you will do in the future, you have already done in the subjective reality around other observers.
Whether the above assertions have anything to do with reality or if they only exist in our model of reality, is up to debate.
What I would like to say about the model of 4D block time is that while it is understandable how at any arbitrary moment of our worldline the state of our brains could only have memories of the past and semantical predictions of the future - and in this sense it could be said that time is only an illusion - I personally find it philosophically very very difficult view as far as considerations of reality goes, because then there is absolutely no meaning or reality to our conscious experience where we experience as if time exists in one state "at a time" and flows forwards.
I find it practically impossible to wrap your mind around the sort of reality where "this" moment is not truly occurring, but rather all the so-called "moments" exists at once and everything is still. I find it far more likely that block time model of reality is simply too much of an approximation to be able to address these issues properly, and bitw, this is also where the philosophy of the mind needs to be taken into consideration; another not-so-light subject of metaphysics :)
I wish I had more time to communicate these things more clearly, but I hope you can pick up just what a huge issue this is for anyone considering the metaphysical nature of reality.
Let it be still said that if we accept the framework of SR, I personally find it most likely that there is a subjective reality around us where there is a subjective "now"-moment in which things may move forwards and backwards when we change direction, as described by the "horizontal line" of Minkowsky diagram. You can still "visualize" this as if there is a block-time or whatever, as long as you also imagine there really are 3D slices flowing through the block (in different orientations) in a fundamental sense. This is because nothing can deny the fact that our subjective reality *does* consist of time flowing forwards, not a static block of events.
And as always, reality is painfully, painfully elusive...
