Is there a traditionally accepted threshold probability at which highly unlikely becomes impossible? Many events which by any practical definition are impossible have extremely low but nonzero probability of occurrence. For instance, the positions of oxygen molecules in a room are basically random and independent, but it is clearly impossible that they all end up in one half of the room at the same time. Is there a standard order of magnitude of probability where the transition from unlikely to impossible occurs? In terms of coin flips, I don’t think 50 consecutive heads would be considered impossible, but I guess that 500 consecutive would. 
I was hoping for a general answer, application independent, but it seems the consensus is that there isn’t one. 
 A: 
Is there a traditionally accepted threshold probability at which highly unlikely becomes impossible?

No, there isn't. People will in practice set certain probability thresholds that are specific to certain applications. For example, suppose a drug company is doing an automated trial of 10,000 new substances in vitro as possible drugs for killing antibiotic-resistant bacteria. Then they probably will not follow up on one of these substances unless it shows an effect that would have a probability $\lesssim 10^{-4}$ of occurring by chance. Setting a less stringent threshold would essentially guarantee that they would be doing follow-ups on a whole bunch of substances that don't work.
A: Reference classes
No, there is no standard finite probability that is used as a threshold for impossibility. This is largely because it depends on what your "reference class" is for the process that you're studying, and how often things in that reference class are happening.
So if you were talking about human-scale events happening within your lifetime, your lifetime is on the half-order of $10^{9}\text{ s}$ (that is, generally between 10 and 100 years, this number of seconds being about 30 years). If one thing happens to you every second or every tenth of a second or so, you might consider, say, a probability of $10^{-12}$ to be "impossible" for such things. But this definition does not nicely transfer to other events. If my computer is generating individual random events about as fast as one of its cores can generate random numbers, it might hit one of those unlikely events in 1-2 hours, perhaps. So $10^{10}$ things may happen to you in your entire lifetime, but $10^{11}$ things easily happen to my computer per hour, and however many things can happen to a system determines what probability is practically "impossible." 
Reference-class independent probabilities due to energy limits
There exists an upper limit on this: there is an upper limit on the frequency that things can happen before they start to generate black holes, and a limit on the size of the observable universe. Seth Lloyd at MIT calculates it at $10^{120}$ operations total. One can improve this a little bit by confining the issue only within our solar system rather than measuring the mass of the entire observable universe, and bracketing it to events that only happen once every twenty thousand years or so, but it doesn't reduce the exponent by that much: one still has a formidable exponent of $$\frac{M_\text{Sun} c^2}{\hbar}~T\approx10^{93}.$$But, please notice that because these are discussing the information processing capacity of raw mass-energy, for any real practical purpose the threshold probabilities are going to be much, much higher than this $10^{-93}$ probability estimate would suggest.
Computational probabilities
I only know of one academic field where there is some semblance of agreement about what is complex, and that is cryptography. Cryptographers overwhelmingly use bits to measure complexity and so I will be using base-2 rather than base-10, but there is a nice rule of thumb that $2^{10} = 1024 \approx 1000 = 10^{3}$ so you can very often just multiply by about 0.3 to get the relevant base-10 exponent. 
In computing, is known that the present largest supercomputer in the world (as of 2018, IBM's Summit, installed at the US Department of Energy's Oak Ridge National Laboratory) does approximately $2^{82}$ computations per year. This means that cryptographic attacks with a probability of $2^{-80}\approx 10^{-24}$ may be presently possible for some extremely high-value data, and that exponent is not "too far off" for much less valuable data -- for example the bottom of the top-500 list usually trails this performance by about $2^{8}$ and renting out for a few weeks rather than a year saves you about $2^{4}$ in cost, so maybe you can count yourself reasonably safe against things that happen with probability $2^{-70}\approx 10^{-21}$. On the other hand, maybe not: we know that supercomputer performance has been increasing by maybe a factor of 2 per year, so that exponent ticks up by one every single year.
Based on that sort of scale, most cryptographers are willing to tell you that an ideal block cipher should probably have at least a 128-bit key if it needs to secure things for the next 50ish years before you upgrade to a new infrastructure; on the other hand they will typically recommend a 256-bit hash function. That one of these is double the other has to do with a square root: typically with hash functions one is concerned with coincidences between two independent values, and often coincidences scale with the number of "handshakes" in a set of size $n$, which scales like $n^2/2.$  (This is sometimes known as the "birthday paradox", after the surprise that most people have at finding that even if birthdays are allocated in a completely uniform random way, there's a 63% chance that a room of only $n=\sqrt{2\cdot 365}\approx 27$ people has at least one shared birthday between them.)
A: This idea that events with probability under some small value are to be treated as impossible is known as Borel's or Cournot's law, but is not actually an accepted law, as far as I know.
It is however understandable that in applications of probability calculus, the events with extremely low probability (such as Sun exploding nova or gas in a room collapsing in one of its corners) are routinely ignored (thus treated as if impossible).
However, the value of probability under which events should be so ignored depends on the question that we try to answer. Sometimes such simplification is not possible. As is often said, extremely improbable events happen all the time.
Borel's paper Sur les probabilites universellement negligeables can be read here:
https://gallica.bnf.fr/ark:/12148/bpt6k3143v/f539.image
