Is it possible to get a negative F-statistic (e.g. in ANOVA)? If so, why? If not, why not?
I'm thinking of an ANOVA context, but if the answer changes depending on context I would also be interested to know why that is.
 A: Updated to account for the "F statistic" in population genetics that @whuber links to above. 
It depends on what you mean by "F-statistic" and "context".  In statistics, the $F$-statistic is a ratio of variances.  As long as the "context" refers to this F-statistic, whether it comes from an ANOVA or a simple linear regression model will not matter.  Evidently however, there is something unrelated called an "F statistic" in population genetics.  Here is a quote from the linked page:  

F statistics: The F statistics in population genetics has [sic] nothing to do the F statistics evaluating differences in variances. Here F stands for fixation index, fixation being increased homozygosity resulting from inbreeding.  ...  
...The value of FIS ranges between -1 and +1. Negative FIS values indicate heterozygote excess (outbreeding) and positive values indicate heterozygote deficiency (inbreeding) compared with HWE expectations. 


On the other hand, it is not possible to get a negative $F$-statistic of the type that occurs in statistics.  Think about what an $F$-statistic is.  All $F$-statistics are of the form:
$$
F = \frac{MS_{\rm effect}}{MS_{\rm residual}}
$$
So it could only be negative if (only) one of those values were negative.  But all mean squares are of the form:
$$
MS = \frac{SS}{df}
$$
So it, in turn, could only be negative if (only) one of those values were negative.  Now consider the form these must take:
$$
SS = \Sigma_i(x_i - \bar x)^2  \\
\text{and}  \\
df = N - \text{number of parameters estimated}
$$
Squaring any value yields a positive value.  I suppose you could say that if you were estimating more parameters than you had data, then you would get a negative df, and thus ultimately a negative $F$, but such a model would be unidentifiable, so this could not be done.  Thus, any $F$-statistic will always be non-negative.  For a given sample, it is possible to get $0$ if all conditional means are identical, or undefined if all data exactly equal the conditional means, but these are extremely unlikely to happen in practice even if the null hypothesis is completely true.  
