From all of the answers and the comments therein, I think the issue is simply a lack of understanding of relativity, so this answer is mostly just going to be a primer in how to think about it.
I am not a teacher, nor a physicist. Forgive me if I make errors, and point them out so I can correct them. However, I am friends with an excelent physics teacher, and I got the privilege of sitting in on his class when they were learning relativity. He started out with a few of the more interesting thought experiments, like the one with two flashing lights on a train. He kept going and going, and you could watch the students getting more and more perplexed. Finally one student slammed his hand down on the table and yelled "Bull-----!" My friend did not miss a beat, and immediately replied, "Good! Now class can begin!" If the claims of relativity do not bug you at first, that means you are not paying enough attention. They should bug you. They're freaking weird!
Now I am just writing this text as I go. I don't have the privilege of waiting until I hear you shout "BS!" at me, so I have to take a different approach. I find it's effective to make sense of relativity from a historical perspective. The original investigators got to think the same sorts of naive theories that we all think of, and as relativity evolved, they tried to call BS, but couldn't because the evidence showed that the universe was actually that weird.
Before Einstein, there was Maxwell. Maxwell's equations were a very effective set of equations for modeling the propagation of EM waves. They predicted basically everything we could test when it came to EM waves. However, there was a catch. In Maxwell's equations there was a really natural derivation of a "wave velocity," that is the velocity that an EM wave propagates in. Now if we do any of the standard experiments where I stand on a train and throw a baseball or something, we're used to the idea that I perceive the baseball as traveling slower than you perceive it when you're on the ground. But Maxwell's equations only offered one wave velocity for EM radiation, c -- the speed of light.
The natural assumption that many scientists made in that age was that light traveled through some sort of medium, just like soundwaves travel through air. Thus the natural "privileged frame" for defining the speed of light was the frame of this medium, called the Aether. And, of course, we expected it to behave similar to how air does, with doppler shifts and shock waves and what-not. There were other theories too, but Aether is the most fun to talk about, and it lines up directly with your concept of an electron traveling at a high velocity with respect to a ship that is traveling at a high velocity.
At some point, scientists started experimenting to try to pin down this Aether's frame. They did many experiments, the most notable was the Michaelson-Moorley experiments. The idea behind these experiments was that the Earth is clearly orbiting the sun, so its velocity with respect to the Aether should change over the course of a year. The experimenters just had to look at the velocity of light waves in a couple directions, and see how they differ. Then we could make calculations about our movement through the Aether.
When they compiled their data, they noticed a really strange thing. The velocity of light was the same in all directions. It didn't matter if you pointed the light with the orbit of the Earth, or against it, or if the Earth was on one side of the sun or the other. The measured speed of light was the same every time. This meant one of a few things:
- The Earth really was the center of the universe, being connected directly to this heavenly Aether.
- The Aether was somehow being "drug" along by the Earth so completely that we couldn't even detect any relative velocity (they couldn't fly experiments in space at that time)
- Maxwell's equations were incomplete.
Eventually it was found that the Aether drag, or whatever other effect was occuring, caused EM waves to exhibit what is known as the Lorentz Transform. They found that if you took Maxwell's equations in one frame and modified them using the following 4 subsitutions:
Where $v$ is the relative velocity between this frame and the next frame in the x axis, you got numbers which lined up. This transform "fixed" the equations so that they matched the data. Science is good about that: always make sure your model matches the data, even if the data looks schizophrenic.
Now I don't think you need to appreciate exactly what those equations are. What you have to appreciate is that they are messy. Physicists hate messy. Messy usually means we didn't understand the problem fully. This transform at least meant our physics wasn't provably wrong, but that doesn't mean the physicists had to like it. This transform has all sort of peculiarities. For example, we see time dilation and length contraction here, written right into the equations for $t^\prime$ and $x^\prime$.
Also note that by twiddling with time, we broke simultaneity. Remember the story my physics teacher friend started with, with the train with flashing lights? It turns out that you and I can disagree on what order lights flash in, based on where we are standing. There may be some "proper time" order for the "correct" observer, but EM didn't provide any way to identify where that "correct" observer was.
Einstein's brilliance was in the idea that all inertial reference frames were the "correct" one. He threw away the idea that simultaneity had any meaning at all, and started from the principle of relativity:
If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.
Okay, maybe Einstein's own words aren't quite so clear to the layman. Let's use a thought experiment with a train. If I'm on the train and you're on the ground, and the train is traveling at a constant speed with respect to the ground, then the laws governing the speed of light are equally valid in both my frame and your frame. I'm allowed to see that the speed of light is $c$ in all directions, and you are allowed to see that the speed of light is $c$ in all directions, even if I am the one holding the flashlight, while traveling really fast on the train.
Sound messed up? Of course it does. Relativity is wierd! But what Einstein proved was that if you start from this principle of relativity, you naturally derive the Lorentz transform from that principle. The horribly ugly math of the Lorentz transform was still valid, but now it had a simple explaination: the laws of physics remain the same while changing frames.
And indeed time dilation was a major factor in the cementing of relativity as "the right theory." Once we had clocks accurate enough to experiment with this, we found that, indeed, this kooky theory was right. Time was not absolute*. Time could progress differently for different frames, and there was no global concept of simultaneity.
So in your relativistic space ship, computers work just like normal, because in their frame, light is traveling at light speed, so electric waves can travel at 0.95c without breaking any speed limits. If those electrons were viewed from the ground, they would be seen as going at 0.9986c, which is faster, but still not faster than light speed. The stretching and contracting of our concepts of time and space would account for all the oddities that arise from that, and if you want to see the equation which you can use to calculate that 0.9986c, I recommend reading el duderino's answer, which is where I stole that number from. His answer also has the equation used to calculate that.
So when you are doing relativistic spaceship thinking, always remember that every frame is a privileged frame for EM radiation. Light travels the same speed no matter what frame you are in, so you can always solve problems by simply picking the most convenient frame. In this case, it's the frame of the space ship.