I’ve managed to find an infinitely long straight line in space that is a total and complete vacuum and will remain so forever. Thanks to the [del]magic[/del] science! of unobtainium, I am ageless, I have an infinite air supply, indestructible space suit and will experience a constant 1G acceleration all the way the speed of light. Sweet. And away I go!
Is there a point where I’m killed by velocity alone?
Not so far as we know. It is rate of acceleration and deceleration that affects things.
And yes for the nitpickers, I know that I “forgot” about food, water, heat, etc. So to paraphrase the MST3K theme song, “If you’re wonder how he eats and breathes and other science facts. Just repeat to yourself, it’s just a question and I should really just relax.” I.e. I was mainly trying to be funny in the OP in setting up the hypothetical. 
My understanding of physics, feeble as it may be, is that you are already traveling at very nearly the speed of light relative to some things in the universe. And so far, you’ve been ticking along just fine. If you happen to encounter any of those things that are moving fast in relation to you, things will get energetic, but, as others have pointed out, that’s due to deceleration.
When you meet a solid object?
It’s not the velocity that kills in this case, it’s the stopicity.
The concept that velocity alone could kill you is now considered quaint. Our current understanding of physics is that constant velocity is basically the same thing as no velocity at all. Both are examples of an inertial frame of reference and there is no way that you can point to one IFR and say that one is somehow more valid or less valid than another IFR.
As for accelerating or spinning, that’s a whole different can of worms.
Not only is there nothing that would kill a human, but there’s also nothing that would register at all on any instrument whatsoever.
That is, unless you run into something else, and there is a bit of a quibble here. Because in the real Universe, you will run into something: Even if there’s no matter on your route, there will still be the cosmic microwave background. And if you go fast enough relative to your local CMB frame, the microwaves in front of you will be blueshifted up into something dangerous.
In fact, we believe that an effect along these lines puts a limit on how fast cosmic ray particles can move. Above what’s called the GZK limit, CMB photons would be blueshifted so high that they’d hit cosmic rays hard enough to (at least temporarily) turn them into other particles, and brake them in the process.
Which, of course, makes the fact that we do see cosmic rays at higher energies than that something of a mystery.
Nice poster/OP combo there, Mr. Roadrunner…
My basic understanding of physics made me suspect that this was the answer. I thought maybe at some high speed there was a possibility that the biological processes might not be able to work because things get strange when you get to high velocities. Thanks for the replies!
You are probably visualizing accurately in a way. If you travel too fast within an atmosphere, so that you are running into gasses and other things which are standing relatively still, it would be the same as running into anything. It still wouldn’t be the fact that you are in motion, which would cause your problems, it would be the fact that what you are running into is not moving at the same speed. So yes, if you are say, in a rocket car on the ground, with no top on the car, so that you were exposed to all the air the car was rocketing through, you’d suffer a lot of damage. There would be a limit to how fast you could travel unprotected through another substance.
Let’s say that you are going headfirst at c-.9V[sub]blood in your veins[/sub]. Wouldn’t you then die since your blood can’t go c+ to get back to your brain?
No, because velocities are not additive in the way that you’re thinking.
So you wouldn’t die? Or you would?
From your point of view, your blood would be traveling at whatever speed it usually does, and so you’d be fine. From any point of view at all, your blood (and everything else about you) would still be traveling at less than c. If this appears to be paradoxical, it’s only because you’re assuming that velocities are linearly additive (which is a very good approximation for speeds much less than c, which is why your intuition developed that way).
Is that where the term “blood bloods” came from? The blood is blue-shifted from moving too fast? 
My feeble understanding says the same. There is no such thing as ‘velocity’ except in relation to something else.
So if the CMB is blue-shifted, and the opposite side red-shifted, shouldn’t that mean we can tell which direction we’re traveling in the universe right now? I thought the CMB was relatively uniform around Earth. Are we not going fast enough?
Yes. That’s exactly where that term came from.
Any local motion will show up as a shift in the CMB. That includes the rotation of the Earth (or satellite around the Earth), the Earth’s orbit of the Sun, the movement of the Sun around the Milky Way, and the movement of the Milky Way towards a certain area known as the Great Attractor. But these are all local on a universal scale, even the last one. They have to be subtracted out of the CMB to see that it is uniform in all directions.
We can always measure our velocity relative to some convenient big thing. That convenient big thing might be the train we’re riding on, or the Earth, or the Sun, or the center of the Galaxy. Or if you want to go really big, that convenient big thing might be the entire portion of the cosmic ionization barrier visible from our location. Look at the red/blue shift of the CMB, and that’s what you’re measuring your velocity relative to. But it’s still just one more in a long list of big things.