What does it mean if the F statistic is zero?

Answer 1

In a simple, one-way ANOVA, the F-statistic is almost never zero, because it is the result of dividing a mean square (regression) by another mean square (error), both of which are usually nonzero, positive numbers.

In very unusual circumstances, if the regression mean square (MSR) is zero, then you could have an F-statistic of zero. For the regression mean square to be zero, your model would have to be a perfect fit of the data, which would indicate severe overfitting of the data.