What's a reasonable mean squared error (or RMSE)? For a given regression dataset, with target $Y \in \mathbb{R}$, where the training set examples have $\mathrm{std}(Y)$ and $\mathrm{mean}(Y)$, what is a mean squared error that makes the result obvious or not interesting? What would be a "reasonable MSE"? (all in relation to the standard deviation and the mean of $Y$).
For example, it doesn't make much sense for such MSE to be larger than the standard deviation squared (variance), because choosing a constant target to be the mean would give such error?

You partially answered yourself. The best simple (single parameter) model that you can have to minimize squared error is to always predict the mean. In such a case, the mean squared error would be equal to the variance. What follows, if your solution is worse than this, it's really bad. Other than that, MSE or RMSE are not bounded, they can take any non-negative values. The only way to judge if the value of the metric is reasonable is by comparing it to some benchmark, like the results of another model.
