How large does $N$ need to be for statistical mechanics to be a good approximation? About how many components ($N$) does a system need for statistical mechanics to apply to that system?
I took stat mech and biophysics from the same professor in undergrad and I distinctly remember him saying that part of the reason that biophysics was so intractable is because systems were large, but not so large that the thermodynamic limit made sense. I think he said biophysical systems often had $N \sim 10^2-10^4$ components while stat mech really only made sense for systems with $N>10^6$(?), but I really don't remember the exact values for $N$.
 A: I think it's unfair to ask for an exact value for $N$ to justify all statistical mechanics. There are very many different problems and applications of stat mech, and some of them might have intrinsically low variance and work fine for relatively small $N$, whereas in other problems really require $N \rightarrow \infty$.
With that said, it's easy to see why statistical mechanics works so much better for (say) a gas of particles than for many biological systems. As you mention in your post, biological systems often deal with $N \sim 100$ or $1,000$ degrees of freedom, which is firmly in the "mesoscopic" regime. On the other hand, a reasonably sized box of air will contain something on the order of a mole of particles, ie $6\times 10^{23}$ particles, which is twenty orders of magnitude larger than the biological system. So you can see why you typically don't need to scratch your head over whether $10^6$ or $10^7$ particles is enough to justify stat mech: typically, the number of particles is so unimaginably huge that you can often (but not always!) approximate the system by taking the thermodynamic limit $N \rightarrow \infty$.
Another nice semi-quantitative heuristic: very often one approximates particles as non-interacting in statistical mechanics. In this case, many observable quantities are given by a sum of independent and identically distributed random variables, such that the sum can be approximated to good precision as Gaussian distributed by the central limit theorem. Then, you expect that the standard deviation of the distribution will fall as $1/\sqrt{N}$ as $N$ becomes large. From this, you can get a rough idea of how good statistical mechanics will be as $N$ increases: for instance, $1/\sqrt{100}$ is $0.1$ for common biological systems, while $1/\sqrt{10^{23}}$ is $3 \times 10^{-12}$ for a box of gas particles.
A: The quality of an approximation being "good" is not binary. There is no point at which an approximation switches from being good to being bad. Rather, its a continuum. The larger the $N$, the better the approximation becomes. There is no fixed $N$ for which statistical mechanics suddenly kicks in.
This reminds me of an old lecture by Planck ("Atomic Theory of Matter" in "Eight Lectures on Theoretical Physics"), which I hope OP will find useful. You can find it online. He goes:

How many atoms are at least necessary in order that a process
may be considered irreversible? the answer is: so many atoms that one may form from them definite mean values which define the state in a macroscopic sense.

Planck will not give you a specific number. How could he? If he did, the obvious question would arise: does statistical mechanics break down if you take $N-1$ atoms instead of $N$? No? Then how about $N-2$? No again? Then ... one eternity later... then how about $N-N$? Oh, now it does? But when did it change?
A: One of the problems comes from the Stirling approximation for the factorial in the Boltzmann distribution.
When deriving the Boltzmann distribution, there is a step where $\log (N!)$ is approximated by $N\log(N) - N$. The approximation is used for the total $N$, and also for the number of occupied states $N_k$ in a given level of energy $E_k$.
The approximation is better for larger $N$'s. An error of $1\%$ for $N=10^2$ and $0.01\%$ for $N = 10^4$.
When $N$ is not huge enough, its is possible that several energy levels have
their $N_k$ below 100 for example, where the approximation starts to fail.
A: I think some information should be added to the previous answers, which contain good general points but miss two more specific issues relevant for answering your question.
The question about how large has to be $N$ for using statistical mechanics with confidence certainly doesn't have a sharp number as an answer, and indeed, the use of statistical mechanics for the ordinary matter in a lab is justified by numbers of degrees of freedom of the order of Avogadro number.
However, experiments on small atomic clusters and more than half a century of computer simulation have shown that the real answer may be more subtle.
First of all, we have to distinguish between the values of $N$ providing a good approximation for the thermodynamic limit ($N\rightarrow \infty$) and the values of $N$ enabling us to use statistical methods safely.
Indeed, the behavior of systems as small as a few tens of atoms may provide a satisfactory qualitative and quantitative description of thermodynamic properties almost everywhere in the phase diagram, provided the effect of boundary surfaces is minimized (for example, through periodic boundary conditions). The key lesson is that an important parameter is the correlation length. The size of the system has to be larger than the correlation length $\xi$. Except for continuous phase transitions, the number of degrees of freedom inside a volume $\xi^D$, where $D$ is the dimensionality of the space, is not huge and much below the Avogadro number.
We have also to recall that a statistical description of the behavior of small systems is possible independently of the problem of approximating the behavior of bulk systems. In that case, we can continue using the usual ensembles, ut we need to choose the most adapted to the physical conditions.
A second point, specific to gravitational systems, is that application of statistical mechanics methods should take into account the peculiarities of long-range gravitational systems (non-extensiveness of the energy and entropy and negative specific heat). The presence of such oddities does not hamper the use of statistical mechanics methods but requires a more careful analysis than usual. Also in such a case, the usual equivalence of the ensembles at the thermodynamic limit is lost. Also, the usual way of analyzing the size dependence of fluctuations requires a deep revision. Still, a statistical mechanics description of stellar systems can be done safely even with numbers much below the Avogadro number.
