Nice post from Daniel Lakens.
The table shows the bias. With four groups of n = 20, a One-Way ANOVA with a medium effect (true η² = 0.0588) will overestimate the true effect size on average by 0.03512, for an average observed η² of = 0.0588 + 0.0347 = 0.0935. We can see that for small effects (η² = 0.0099) the bias is actually larger than the true effect size (up to ANOVA’s with 70 participants in each condition).
When there is no true effect, η² from small studies can easily give the wrong impression that there is a real small to medium effect, just due to the bias. Your p-value would not be statistically significant, but this overestimation could be problematic if you ignore the p-value and just focus on estimation.