Is $R^2$ useful or dangerous?

I was skimming through some lecture notes by Cosma Shalizi (in particular, section 2.1.1 of the second lecture ), and was reminded that you can get very low $R^2$ even when you have a completely linear model. To paraphrase Shalizi's example: suppose you have a model $Y = aX + \epsilon$, where $a$ is known. Then $\newcommand{\Var}{\mathrm{Var}}\Var[Y] = a^2 \Var[x] + \Var[\epsilon]$ and the amount of explained variance is $a^2 \Var[X]$, so $R^2 = \frac{a^2 \Var[x]}{a^2 \Var[X] + \Var[\epsilon]}$. This goes to 0 as $\Var[X] \rightarrow 0$ and to 1 as $\Var[X] \rightarrow \infty$. Conversely, you can get high $R^2$ even when your model is noticeably non-linear. (Anyone have a good example offhand?) So when is $R^2$ a useful statistic, and when should it be ignored?