Imagine a light spaceship capable of traveling at 0.999c is tasked to tow a radioactive cargo containing millions of tons of polonium and radium, however the container holding the radioactive materials made of 100mm thick concrete with 50mm lead layer tethered by a 10000mm long carbon nanotube cable, my concern is the close proximity between my crews and the cargo due to length contraction when going 0.999c? Am I being paranoid or I need to cancel the shipment as I don't want to compromise safety?
Length contraction, like time dilation, is only seen by outside observers. To the crew, their spaceship & cargo, and their perception of time, will appear unchanged. It's only the outside universe that will look weird.
So to the crew, the cargo will always appear to be at the end of a tether that's 10,000 mm* long.
*Did you really mean 10,000 millimeters? That's only 10 meters. 10,000 meters would be more reasonable.
In any reasonable situation, the trip is absolutely pointless in the extreme. Totally counterproductive, even directly harmful.
At 0.999 light speed, the gamma factor is 22.37.
That means, in order to accelerate a mass to 0.999 light speed you must expend an energy 21.37 times the mass-energy equivalent. That is, you would need to expend grotesquely more energy than you could, by any means, get out of the material you are transporting. Where do you get that energy?
In addition, you must deal with the rocket equation. The reaction mass you must expend is many times the mass you must transport. Since you must accelerate at least part of your reaction mass as well, the rocket equation is a real killer here. You wind up needing to expend many thousands of times the total "payload" as reaction mass and as fuel to get the energy.
So, imagine you have a monstrous large supply of fusion fuel. Many thousands of times the mass you are disposing of. (Tens of thousands? I'd have to do some difficult math to be more precise.) And you run a fusion rocket to get power for your spaceship. And you pump out several thousand times the total mass of fusion products as reaction mass. That material winds up as a nasty backwash from your rocket motors.
So, if you are trying to get rid of Radium and Polonium, it seems kind of silly to try to do so by creating thousands of times as much mass as blazing hot (both temperature and radioactivity) rocket exhaust, and shooting that directly back towards the place you are leaving. People might complain.
In addition, Polonium and Radium are pretty easy to store. They are alpha and beta emitters, so they are not really a problem unless you eat them. You just take it to some unused desert without much ground water flow, and you leave it there. The very fact it is radioactive means it is decreasing the problem nature on its own. The longest half life isotope of Polonium is Po209 with a half life of 125.2 years. So after that long, your problem is half as big. If you really have a phobia about radioactivity, you just park it on some unused part of the moon. No wind, no water, no stray animals. It stays there until you come back for it.
James' answer is fundamentally correct.
On top of that, consider that with time contraction you also have time dilation, so for an external observer the time between nuclear decays will be longer, thus the radioactive fuel will be less radioactive, "magically" matching the shorter shielding.
If you are flexible on the shielding materials, this trip is no problem.
While concrete and lead are effective shielding, so too is water. Within just a few meters, the radiation from spent fuel rods can be rendered effectively harmless.
For the kinds of radiation coming off spent nuclear fuel, every 7 centimeters of water cuts the amount of radiation in half.
At several meters in depth, you could be so bold as to swim in the pool if you felt so inclined. Although water tends to be heavy/lots of mass, since you are hauling millions of tons this doesn't seem like much of a stretch.
A final point is that the concrete may in some circumstances, be more dangerous to your crew than the fuel rods. It will absorb oxygen in the room, which is not so bad on Earth but a potential serious problem on a spaceship. Any repairs to it will also be a problem for the same reason. Another issue is that it can also absorb toxins so it will be radiating those into the environment around it as well.
A similar real life example of this is a crew that entered a ship compartment that was closed for a extended period of time. Steel naturally oxidizes and rusts in the presence of oxygen. This rust built up in the compartment to such a high concentration that in the span of time it took the unfortunate crew member to climb the ladder down into the compartment and then start his climb back up, he was rendered unconscious. This is just one example that poor ventilation in a closed environment like a spaceship can be lethal.
So, my recommendations are to:
Increase the amount of shielding, either metal, water, etc.
Slim down the size and mass of the ship and send it unmanned and collect it at the destination or use some sort of AI autopilot.