Is it possible to break the second law of thermodynamics regarding entropy? The motivation behind my question is that it seems very unlikely that a chunk of metal would "randomly" reach escape velocity and fly away from the Earth, but it happens thanks to NASA and other space programs. If the second law of thermodynamics regarding entropy is statistical, can it be broken on a larger scale?
 A: Entropy may decrease locally, but if you look at the complete system, entropy is strictly non-decreasing. Living organisms are an excellent example of a subsystem that produces order from the environment, organizing molecules into fantastically complex structures. But although they can reduce entropy locally, they must increase the entropy of their surroundings by at least as much in the process.
So, in a sense, your intuition is actually opposite to the truth - you can decrease entropy of a non-isolated system on a small scale, but at a large enough scale that includes all components in the system, entropy never decreases. Anything that seems to result in a decrease in entropy must increase entropy somewhere else.
A: Note: Was going to comment on @Nuclear Wang's answer, but Without the 50 reputation yet, I'd like to build off their example of living organisms being a local system with a decrease in entropy, and how this is balanced out.
As @Nuclear Wang pointed out about living organisms:

But although they can reduce entropy locally, they must increase the entropy of their surroundings by at least as much in the process.

This is the crux of your answer. Although a local reduction of entropy does not increase the entropy of it's surroundings, but rather the constant reduction of entropy in the surroundings enable these local low entropy states to exist in order to approach equilibrium. It is crucial distinction to make, between a region of lower relative energy vs. a region who's entropy is constantly being reduced.
The solar system is a great example for visualization purposes, although it is itself a subsystem of a larger system, ad infinitum.
In order for any life on Earth to exist in a manner that reduces entropy; so much so that 7+ billion humans can exist, build cities that remain rigid, and coexist with trillions of other organisms, somewhere outside of the system there must be a region of spacetime with a very high entropy interacting with this system. The sun, a region of very high energy, directly interacts with the Earth. There are no true closed systems.
Aside
I am currently working on a personal theory (inspired by the work of Jarzynski and Crook) that the second law of thermodynamics is the most fundamentally important reason that life exists; that the formation of amino acids, and complex life is a necessary byproduct in every solar system (with the proper precursors) in order to approach equilibrium.
Any solar system as an isolated system (each star is so far apart from the next, it's isolated enough) and entropy cannot decrease over time, even as the entropy of the star is reducing through nuclear fusion. As energy from the star is added to say, the ocean of a planet, the entropy increases locally before being dissipated through the medium. Along the way, areas of uneven energy cause molecules and atoms to organize themselves in a manner which is conducive to (a) the medium of energy, and (b) the manner in which entropy equilibrium is reached (the environment). These reorganized molecules would themselves be local areas of low entropy, further altering the manner in which entropy/energy is dissipated through the system.
A: Yes, it is, but it is, indeed, as you say, "extremely unlikely" in a very specific sense. To the best of our knowledge, there is nothing at all to prevent the atoms and molecules in the Earth from all moving to conspire "just right" so that at a particular point in time, they give it a huge "kick" that sends it reeling. Such movements are entirely possible thanks to the time-reversal symmetry of the laws of physics. Moreover, so long as their movements are truly and thoroughly random (that is, enough to make the system "ergodic"), there will always be a nonzero probability that by pure luck a configuration arbitrarily close to this will happen, and that probability for such will approach 1 given unlimited time.
This is for the same reason that you can, from a perfectly fair coin, get an arbitrarily long sequence of "heads" or "tails", e.g. you could flip a truly fair coin and get "heads" 100,000 times in a row, at least, so long as you flip long enough.
The catch, though, of course, is just what that nonzero probability is or, equivalently, how long you have to flip the coin for: for the coin, it's 1 chance in $2^{100\thinspace000}$, and for the thermodynamic effect, it's stupidly more orders of magnitude still (exponents on the order of Avogadro's number or more, I believe). Hence, we will far-more-than-as-sure-as-you-won't-win-the-lottery-a-billion-times almost surely never witness such an event - it is so stupidly unlikely that were it to actually happen, we'd have good reason to more likely wonder if we got something wrong about physics, or perhaps that ghosts or some other such extra-physical thing actually exist, or perhaps that the Universe is a simulation and it had a glitch, or any of a number of other such possibilities; just as if our coin actually did come up 100,000 times "heads" we would highly likely think it must be biased. That is, we would be more likely to, indeed, suspect a genuine breach of the second law, before concluding that, in fact, the second law had operated as we've thought.
And, moreover, there is no way to arrange any matter, using only matter within the Universe, to cause this on purpose, without just-as-almost-surely increasing entropy elsewhere; because that's then a process involving the second law - you just have to broaden the context. This is how NASA's rockets, as you mention, work: the burnt rocket fuel represents a much higher entropy than the reduction required to create the ordered motion away from the Earth.
A: The phrasing of the question confuses physical laws with human laws. Human laws are prescriptive: they tell you what to do, and if you don't comply, you are breaking the law. Physical laws are descriptive, they describe the way things are. If something doesn't behave the way a physical law says they should, that's not breaking the law: it means the law is not actually a law at all.
What this question is actually concerned with is identifying the class of systems to which the second law of thermodynamics is applicable. But having restated the question, I'll leave answering it to people whose knowledge of physics is less rusty than mine.
