F-ratio Question What does an F-ratio of 0.00 tell us about the two predictors (both quantitative) in a regression analysis?
and
What does an F-ratio of 0.00 tellus about the group means in ANOVA?
 A: The F-ratio is the variance attributable to the model divided by the variance in the residuals. In the case of an ANOVA, where the whole model is just group identities, that translates to the variance between groups divided by the variance within groups --- but the more general form of the definition is applicable to both regressions and ANOVAs (and everything in between).
An F-ratio of 0 means that either there is no variance at all attributable to the model (i.e. all variance in the outcome is random error) or that the error variance is so overwhelmingly huge that the ratio is tiny enough that it rounds to 0. In the most typical cases[1], in order to obtain an F-ratio of 0 in a regression with two quantitative predictors both predictors should have no correlation with the outcome. To get an F-ratio of 0 in an ANOVA, the mean of each group should be identical (there is no variance between groups). 
[1] This is not the only possible explanation, though. See whuber's comment for an example of a completely different scenario that will also yield F=0. 
