In my course notes it's recommended to compute the standard deviation of the residuals in order to infer something regarding the goodness of fit of the linear model.
The notes also say that the lesser this number is, the better the model fits. However, what's small enough number? What's too large number? And what does the number "mean"?
The standard deviation of your model is the $\sqrt{MSE}$ (square root of Mean Squared Error). Basically, you want to look at your R-squared (or Adjusted R-Squared) and your F-statistic.
The notes also say that the lesser this number is, the better the model fits. However, what's small enough number? What's too large number? And what does the number "mean"?
Your MSE is not bounded to a simple interpretable scale, so using the R-squared is a much more useful tool. Generally, a smaller $\sqrt{MSE}$, the better, but like you imply, there's no way to really know how small or large the measure could be. Since the R-squared is bounded between 0 and 1, you can judge how large the variance is by how small the R-squared is. The R-squared is also a measure of effect size, which you should always report. And the F-statistic give you and overall goodness-of-fit statistic. For your overall model, your F-statistic is calculated by $\frac{MSR}{MSE}$.
I'll assume you do not have notes on how the R-squared or F-Statistic are calculated, so here is a reference. Please refer to pages 5 and 14. http://www.stat.ufl.edu/~winner/statnotescomp/regression.pdf
And here is a reference for interpretation: http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-do-i-interpret-r-squared-and-assess-the-goodness-of-fit
Hope this helps
I'll add to the other answer by saying that the residual standard deviation is simply: $$s_{res} = \sqrt{\frac{\sum_{i=1}^{n}{(\hat{Y}_{i} - Y_i)^2}}{n - 1}}$$
Which is the sample standard deviation. If this value is "small enough", then you don't need to add any more terms in your regression model, e.g. $\beta_2y^2 + \beta_3y^3 + ...$ Of course, there is no hard and fast rule for what is "small enough". So typically you'd compare models against each other, and, along with a high adjusted R-squared, pick the model with lower $s_{res}$.
Note that if you keep adding terms, your $s_{res}$ will likely decrease as you get closer to overfitting your data, i.e. making a model that's more complex than needed. What you may consider doing in this case, is making an "elbow" plot. Basically you plot all of your models from least complex to most, and notice how the $s_{res}$ decreases as model complexity increases. At some point, you'll see the $s_{res}$ leveling off, and you can use that as an indication of where to stop adding terms to your model.
