More concretely, imagine Alice and Bob are moving away from each other at 50% the speed of light. They both observe event some event, X, occur and note the location in space-time. If Alice observes that X occurs in the location (t,x,y,z), then we can compute precisely where and when Bob observed the event occurring (t',x',y',z'). This conversion is known as the Lorentz Transformation , and can be derived mathematically from the assumption that the speed of light is constant regardless of reference frame.
Where this gets weird is the case where Alice and Bob observe 2 events: X and Y. In this case, it is possible that they will disagree about which event happened first. Once you accept this, it should become clear why causality requires some speed limit. You can do the math based on the Lorentz Transform and confirm that this limit is the speed of light. Intuitively, this is a direct consequence of the fact that we defined the Lorentz Transform to make the speed of light constant.
In case the need for a speed limit is not obvious, let as pretend that it does not exist. Suppose that, from Alice's perspective, X happened before Y. Further, suppose that Carol happened to be in a spaceship that passed by X at the instant it occurred, moving at a velocity that would take her by Y at the instant it occurred.  From the perspective of Bob, Carol would have traveled backwards, going from Y to X.
We can make this situation even worse by considering Carol's perspective. Recall that we defined Carol as starting at X and traveling to Y. In the same way, we can define Dave as starting at Y and traveling to X (recall that, if we were Bob, we would be convinced that Y happened first, so with a fast enough ship, Dave and make it in time). In this situation, both Carol and Dave exist at Y, so Carol can give Dave a copy of her diary of the trip. However, after Dave makes the trip to X, he will meet Carol again, so he can give her a copy of her diary of the trip she is about to take.
 This means that if I am on Earth, and you are in a space-ship moving at 99% the speed of light (from my perspective), and we both measure the speed of light, we will arrive at the same answer.
 This is possible only because there is no limit to how fast Carol's Spaceship can travel.
So what? It's Bobs problem.
FTL electrons, in medium where speed of light is much less than c, are traveling exactly as you described.
In other words, as Carol travels 'forward' in Bob's time, her clock runs backwards. Or, from Carol's perspective, Bob's clock would be running backwards.
This isn't actually a problem in relativity, but is the definition of time travel.
However, in the four person example, Dave is able to hand Carol her diary of the trip that she is about to make. If that is not time travel, I do not know what is.
The only reason that this cannot happen is that X and Y are so far away in space, and close in time, that it is impossible to travel between them.
This is commonly illustrated with the following scenario. Imagine Alice is sitting by the train tracks. Her friend Bob comes past riding on top of a train, sitting exactly in the middle of the car's roof. As he passes Alice, they high-five (presumably Alice is on a raised platform of some sort). A split second later, Alice sees lightning strike each end of Bob's carriage at exactly the same moment. Thanks to some very precise measuring equipment, Alice is able to determine that the two lightning strikes occurred at the very moment that she high-fived Bob. At that point, Alice was exactly halfway between the two points that were struck, so it makes sense that the light from those strikes reaches her at exactly the same time.
Alice also knows that Bob would have seen lightning strike the front of the car before it struck the back of it: the light from the front strike would have passed Bob on its way to Alice, and the light from the rear strike would have passed Alice on its way to Bob.
Now let's look at it from Bob's view. As Alice concluded, he sees the lightning strike the front of the car first. But (a) he's exactly the same distance between the two strike-points, and (b) the speed of light is always the same for any observer. So if he sees the light from the strike at the front first, that means the front was struck first. Bob has a different order of events from Alice.
Which order is the "right" one? Answer: both. Or, if you prefer, neither. There are no grounds to prefer Alice's view over Bob's, or vice versa. You cannot say that one lightning strike "really" happened first, or that they "really" happened simultaneously.
To complete the picture, let's imagine Charlie riding another train car on the set of tracks the other side of Bob's, travelling in the opposite direction to Bob. At the exact moment Alice is high-fiving Bob, Charlie is also exactly lined up with them and smacks Bob on the back of the head. For reasons similar to but opposite to Bob, Charlie will first see lightning strike the rear of Bob's car and then the front, which means that in his (equally valid) frame of reference, the rear strike happened first.
Now imagine someone or something used FTL travel to go from the front of Bob's car to the rear, leaving at the moment the front was struck by lightning and arriving at the moment the rear was struck. In Bob's frame, this would be unremarkable, except for the exceptional speed (ignoring for the moment any adverse environmental effects -- Google "what-if xkcd relativistic baseball" for a flavour of what those might be). But in Charlie's frame, this would be travel backwards in time, as the arrival at the rear would occur before the departure from the front.