Is there any rule of thumb to classify $R^2$ as small, medium or large effect size? I have seen authors saying $R^2 = .13$ is moderate, but they don't mention any source.
 A: The reference you are looking for comes from the behavioral sciences. Cohen (1988) proposed 'small', 'medium', and 'large' magnitudes for $R^2$, standardized mean differences (Cohen's d), and bivariate correlations (Cohen's r), among other measures. The proposed values obviously don't come from thin air, there is a justification for them, but Cohen himself explains these are just very general definitions not set in stone and that specific subject matter also weighs in determining what a relevant effect size is. Specifically for $R^2$, as per pp. 413-414 of the book, the proposed 'small', 'medium' and 'large' values are 0.02, 0.13, and 0.26, respectively. 
Reference:
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd Ed. Hillsdale, NJ: Laurence Erlbaum Associates
A: It depends on the context. 
Sometimes a process is pretty deterministic and signal is strong relative to noise. Then even a relatively simple benchmark model will have an $R^2$ as high as, say, 0.80. An example could be modelling people's weight given their height, age and gender. 
Other times there is not that much signal and there is a lot of randomness. Then even a sophisticated model may not achieve an $R^2$ as low as, say, 0.20. An example could be modelling the daily movements of stock prices given the prices' own histories (and maybe some other variables).
So it really depends.
A: I agree with Richard Hardy but would like to add that it also depends on the penalty or cost of an error in the context of your model. For example, a model with lower $R^2$ may have less costly errors for your purposes than a model with higher $R^2$ but more costly errors. In other words, $R^2$ needs to be considered in the context of what you are trying to explain with your model.
