Variance estimation for small sample size The following variance estimator of a set of data points $x = (x_1, ..., x_N)$
$$
\text{Var}\,(x) = \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2
$$
has itself a large variance when $N$ is small (in my case $N<10$). Looking around for better estimators, I found the median absolute deviation (MAD) that seems to be similar. However, comparing the var values to the MAD values, the latter are smaller. So, they don't really correspond to a variance estimate.
Are there other variance estimators that are robust against small sample sizes?

In a certain sense (unbiased minimum variance) you can't really do better than the estimator you gave above, you are efficiently using the information available in the data you are analyzing. How you can do better is e.g. if you have prior information by using a Bayesian approach or via shrinkage approaches (either sort of Bayesian shrinking towards a sensible guess, or via a hierarchical model across related quantities e.g. estimating variances for the same variable in different populations or something like that).
