It's easily possible for the standard deviation to exceed the mean with non-negative or strictly positive data
I'd describe the case for your data as the standard deviation being close to the mean (not every value is larger and the ones that are larger are generally close). For non-negative data, it does pretty clearly indicate that the data are skew (for example, the gamma distribution with coefficient of variation =1 would be the exponential distribution, so if the data were gamma, they'd look somewhere near exponential)
However, with that sort of sample size, the ANOVA may not be particularly badly affected by that; the uncertainty in the estimate of pooled variance will be pretty small, so we might consider that between the CLT (for the means) and Slutsky's theorem (for the variance estimate on the denominator), an ANOVA will probably work reasonably well, since you'll have an asymptotic chi-square, for which the ANOVA-F with its large denominator-degrees-of-freedom will be a good approximation. (i.e. it should have reasonable level-robustness, and since the means are not so very far from constant, the power shouldn't be too badly impacted by the heteroskedasticity)
That said, if your study will have a smaller sample size, you may be better off looking at using a different test (perhaps a permutation test, or one more suitable for skewed data perhaps one based on a GLM). The change in test may require a somewhat larger sample size than you'd get for a straight ANOVA.
With the original data you could do a power analysis under a suitable model/analysis. Even in the absence of the original data, one could make more plausible assumptions about the distribution (perhaps a variety of them) and investigate the entire power curve (or, more simply, just the type I error rate and the power at whatever effect size is of interest). A variety of reasonable assumptions could be used, which gives some idea of what power may be achieved under plausible circumstances, and how much larger the sample size might need to be.