Meltdown=
More people need to see The China Syndrome, apparently. 
The China Syndrome, as portrayed in fiction, isn't really a thing either. See above photo. The fuel in Chernobyl #4 did melt through the bottom of the reactor core, but it couldn't get through the floor of the structure.
Supposing it could get through the floor, though, the fuel rods would hit the water table, where the water would flash to steam, sending the fuel rocketing skyward into/through the reactor's containment dome. Stuff a turkey with a couple of onions and drop it in a deep fryer at some point to get an idea of how this would work. In the absence of a containment dome (such as with Chernobyl), the fuel would disperse, as sent skyward, creating a much greater fallout hazard than simply exposing the fuel to atmosphere. This would start and be worst nearest the melted-down reactor, though, not the opposite side of the planet.
Removing the water table from the equation, the spent fuel still doesn't get to the other side of the planet. As the fuel travels through the planet, to the core, it will experience dissipative forces of friction and viscosity. In other words, the act of travelling through the Earth's crust, mantle, and core results in a significant portion of the reactor fuel's gravitational potential energy being converted into (more) heat, instead of kinetic energy. Because the kinetic energy of the fuel doesn't equal its original gravitational potential energy, once it reaches the center of the Earth (assuming that the dissipative forces aren't adequate to counter gravity, which is
another leap that we're making for the sake of argument), it will be unable to continue anywhere near as far in the other direction, before gravity reverses its direction to return the fuel to Earth's center again.
Okay, now let's suppose that you built a tunnel, all the way from your melt-down location to the other side of the planet and that you have the means to keep atmospheric gasses from entering that tunnel. No dissipative forces, right? Wrong! You're now operating under the assumption that the mass distribution of Earth is uniform. Guess what.... Earth isn't a uniformly distributed spherical mass. Some parts are denser than others, and those variations in density are not symetrical (and almost certainly not about the axis defined by your magic tunnel). That means that, at various points during the fuel's transit through the tunnel, it's going to get pulled into the tunnel wall, experience friction, lose energy, and therefore
still won't make it to the other side of the planet.
So, in order for
The China Syndrome to be, in any way, relevant, you have to prep the reactor site as follows:
1) Build the site with a floor that the melted reactor fuel can actually melt through. (Meaning that, not only will the reactor fuel melt the floor, but the fuel cannot be bouyant, in the liquid that the floor becomes.)
2) Pump out
all of the water from local aquifers, and seal them off, so that they cannot be replenished.
3) Build a magic tunnel that can maintain a perfect vaccuum, while open to the atmosphere, and traverses a diameter of the planet.
4)
Rebuild the entire planet, upon which the reactor is built, such that it will be uniformly dense, or symetrically dense, about the axis defined by your magic tunnel.
Or, when informing yourself about possible dangers of nuclear energy, don't turn to fiction.
Incidentally, Mongoose, I know you probably weren't bring that up seriously, but others have. Specifically, it's come up as a counterpoint to how safe modern reactors with proper containment domes are. As a result, I bang out this kind of post almost as a reflex.
