First of all we are confident enough that special relativity is correct that we know we would not need to worry about the speed as such: everyone is travelling all the time at any speed you like less than $c$ relative to something, and we don't all explode or die or anything. We also know how to deal with the related issues around communication. This is all very well-tested physics.
Hitting things
What you would certainly have to deal with is the problem of the spacecraft meeting crud in space (dust, gas) which is more-or-less stationary in the frame of the Sun, say, but very much not stationary in its frame. If you hit something with a mass of $.01\,\mathrm{g}$ then it's going to dump about $5\times 10^7 \,\mathrm{J}$ of energy into you. I make that the equivalent of about $10\,\mathrm{kg}$ of TNT. This is probably a significant problem for any vehicle which intends to travel at a significant fraction of $c$: impacts on spacecraft are already a problem, and they get worse as the square of speed: when the spacecraft itself is travelling at a significant fraction of the speed of light, almost anything it hits will be travelling at a significant fraction of the speed of light. This is as much a problem for interstellar travel as it is for interplanetary travel: there's less stuff, perhaps, but you are out there for longer.
Why this is all absurd
Although the person asking the question seems to have repeatedly changed what they're actually interested in in comments, it seems that they're interested in testing things on Earth for use in interplanetary (not interstellar) travel. In the following two sections I'll show why testing such a thing on Earth is not possible, and why such high speeds are impractical for interplanetery travel.
Since the maximum velocities here are only 1% of the speed of light, I'll do all the calculations without relativistic corrections: there will be some, but they'll be small. I'm also generally rounding to 3 significant figures, although some of the values are probably less accurate than that: if you assume $\approx$ whenever I have $=$ it's probably safe. I haven't quoted values for constants like $c$ & $g$, but they're the usual ones.
Testing on Earth is absurd
In order to test anything like this on Earth we would need to accelerate some massive object to $v_\text{max} = 0.01c$. There are two ways we could do this: in a straight tube, and in some ring.
There's a longest straight tube we can make due to the curvature of the Earth: let's say it has $l = 100\,\text{miles} = 1.61\times 10^5\,\mathrm{m}$. We can easily now compute the smallest acceleration (this is simply the case where acceleration is constant) needed to achieve a given $v_\text{max}$ along such a tube, which is:
$$
a = \begin{cases}
\frac{v^2_\text{max}}{l} &\text{object slows down again}\\
\frac{v^2_\text{max}}{2l} &\text{object allowed to hit end of tube}
\end{cases}
$$
And given $v_\text{max} = 3\times 10^6\,\mathrm{ms^{-1}}$, we get, for the first case, $a = 5.59\times 10^7\,\mathrm{ms^{-2}} = 5.71\times 10^6\,g$, and for the second $a = 2.80\times 10^7\,\mathrm{ms^{-2}} = 2.85\times 10^6\,g$.
For both these cases the experiments would last hundredths of a second.
If the object being accelerated had a mass of $1\,\mathrm{kg}$, the energy required would be $1.5 \times 10^{12}\,\mathrm{J}$: this is about equivalent to 359 tonnes of TNT: it's a rather small nuclear weapon's worth. In the case where the object hit the end of the tube this would all be released at that point.
Experiments like this are unlikely to be useful for testing anything: the conditions are so extreme that, while it might be possible to design electronics to survive the version where the object is slowed down again, such electronics would be utterly unlike anything in a spacecraft.
The second version is to test an object which is being run a round a ring. Well, what's the largest ring we could build on Earth? It's a ring which runs all the way around the equator, with radius $R = 3.38\times 10^6\,\mathrm{m}$. So we can now easily compute the acceleration felt by an object running around such a ring:
$$
\begin{align}
a &= \frac{v^2_\text{max}}{R}\\
&= \frac{9\times 10^{12}}{6.38\times 10^6}\,\mathrm{ms^{-2}}\\
&= 1.41\times 10^6\,\mathrm{ms^{-2}}\\
&= 1.44\times 10^5\,g
\end{align}
$$
So this is better: we now only need to design electronics which can survive hundreds of thousands of gravities. We also have the small problem of building an evacuated tube around the equator and somehow steering some macroscopic object around it (how you do that I have, really, no idea at all: the approach used for charged particles in particle accelerators definitely won't work).
This is the kind of thing the ringworld engineers might consider: it's not the sort of thing that humans are going to be doing any time soon.
And, even if you could do this, it's still completely unrelated to the kind of conditions in a spacecraft: the electronics will be completely different, and the accelerations involved are hundreds of thousands to millions of times greater than what is survivable by a human being.
So even if we could do this sort of experiment on Earth, it would be entirely unuseful for spacecraft design. Some things you can only do in space, and this is one of them.
Velocities this high for interplanetary travel are impractical
So, let's look at what sort of interplanetary travel (as opposed to interstellar) might involve velocities this high. We can use the same equations we used before to compute the minimum acceleration involved for a trip of a given length. I will assume here that at the end of the trip you want to stop, because if you don't want to stop then you're definitely talking about the acceleration phase of interstellar travel. And we can treat all the boring orbital velocities as zero, since they're all very small compared to the velocities we're interested in.
We'll take $v_\text{max} = 3\times 10^6\,\mathrm{ms^{-1}}$ again, and this time $d = 5\,\mathrm{AU} = 7.48\times 10^{11}\,\mathrm{m}$. This is about as far as Jupiter.
$$
\begin{align}
a &= \frac{v^2_\text{max}}{d}\\
&= \frac{9\times 10^{12}}{7.48\times 10^{11}}\,\mathrm{ms^{-2}}\\
&= 12.0\,\mathrm{ms^{-2}}\\
&= 1.23\,g
\end{align}
$$
So for a trip to Jupiter during which you accelerate for half the trip and decelerate for the rest this is habitable, For anything shorter than that it rapidly becomes unsurvivable (a trip to Mars would be something like $5\,g$ all the way, which would almost certainly kill the crew I should think).
Well, Jupiter is somewhere interesting to visit, so this is OK. What sort of spacecraft would we need?
Well the first option is a light sail. Let's say the spacecraft weighs $1000\,\mathrm{kg}$ and the sail is a perfect reflector and weighs nothing. How big does it need to be? The pressure on the sail due to light at the orbital radius of the Earth is $P = 2\times 1360/c\,\mathrm{Pa} = 9.07\times 10^{-6}\,\mathrm{Pa}$.
We're going to Jupiter so we need $a = 12.0\,\mathrm{ms^{-2}}$. $F = ma = PA$ so $A = ma/P$, where $A$ is sail area. Plugging in the numbers we get
$$
\begin{align}
A &= \frac{ma}{P}\\
&= \frac{1000\times 12.0}{9.07\times 10^{-6}}\,\mathrm{m^2}\\
&= 1.32 \times 10^9\,\mathrm{m^2}
\end{align}
$$
If the sail is square, it is about $36\,\mathrm{km}$ on a side. It has to be bigger than this of course, because you want to sustain the acceleration beyond Earth's orbit.
So, maybe this is feasible: it's huge but it's not absurd (better make sure you don't use it to focus the reflected light on anything though: it's about a Trinity test every minute's worth of power).
Except ... you can't stop. We don't have a convenient second Sun so when you reach the half way point and want to turn around, you can't. This thing is the canonical example of the acceleration stage of an interstellar trip. So, not useful for interplanetary travel with speeds this high (light sails are potentially useful for much lower speeds where you can rely on the target object capturing you, unfortunately $0.01c$ is much higher than the escape velocity of any planet).
So what about rockets. Well, now we can use the rocket equation:
$$\Delta v = v_e \ln\left(\frac{m_0}{m_f}\right)$$
or
$$m_0 = m_f e^{\frac{\Delta v}{v_e}}$$
Where
- $m_0$ is the launch mass;
- $m_f$ is the final mass which we will take to be $1000\,\mathrm{kg}$
- $\Delta v$ is the total change in speed, which we'll take to be $c/50 = 6\times 10^6\,\mathrm{ms^{-1}}$, as you want to stop at the destination;
- $v_e$ is the exhaust velocity.
So I'll assume some super-duper ion drive which will:
- have enough thrust (they don't);
- have $v_e = 5\times 10^5\,\mathrm{ms^{-1}}$, about 5 times better than anything I know about.
So now, we can plug these numbers in and we get
$$
\begin{align}
m_0 &= m_f e^{\frac{\Delta v}{v_e}}\\
&= 1000\times e^\frac{6\times 10^6}{5\times 10^5}\,\mathrm{kg}\\
&= 1000 \times e^{12}\,\mathrm{kg}\\
&= 1.63\times 10^8\,\mathrm{kg}\\
&= 163\times 10^3\,\text{tonnes}
\end{align}
$$
This is less than twice the mass of a large aircraft carrier.
And this is for a one-way trip: if you want to come back you need to get that mass to the other end. And that means you need a launch mass of $m_0 = 2.65 \times 10^{13}\,\mathrm{kg}$. This is still light compared to the Moon, fortunately.
There are ideas for drives with higher $v_e$ than this but they are increasingly science-fiction devices. Even a high-thrust ion drive is fairly science-fiction: almost certainly, even if it were possible it would require a fusion reactor to power it, and those don't exist yet, let alone ones we can put in spacecraft. The heat-dissipation problems would also be formiddable.
So no plausible kind of propulsion will achieve these speeds for interplanetary travel: they're only interesting for interstellar travel, currently only with light sails of some kind.
Other confusions
One thing the person asking the question was interested in was guidance: how you steer spacecraft travelling at $v\approx 0.01c$. This is a non-problem: if you have a propulsion system which will accelerate you to this kind of speed, you also have one which will change your course when travelling at this speed. The problem is having a propulsion system which will get you to $0.01c$ in the first place.
Conclusion
Travel at $v\approx 0.01c$ is extremely impractical for interplanetary travel: it's not survivable for short trips and the fuel requirements are absurd for all trips. Testing such a system on Earth is completely absurd. This seems to be a case of too-much-science-fiction combined with no real intuition about just how fast $0.01c$ is, how much energy is required to get to that kind of speed, and the implications of the rocket equation.