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Microbial communities analysis is drawing growing attention due to the rapid development of high-throughput sequencing techniques nowadays. The observed data has the following typical characteristics: it is high-dimensional, compositional (lying in a simplex) and even would be leptokurtic and highly skewed due to the existence of overly abundant taxa, which makes the conventional correlation analysis infeasible to study the co-occurrence and co-exclusion relationship between microbial taxa. In this article, we address the challenges of covariance estimation for this kind of data. Assuming the basis covariance matrix lying in a well-recognized class of sparse covariance matrices, we adopt a proxy matrix known as centered log-ratio covariance matrix in the literature, which is approximately indistinguishable from the real basis covariance matrix as the dimensionality tends to infinity. We construct a Median-of-Means (MOM) estimator for the centered log-ratio covariance matrix and propose a thresholding procedure that is adaptive to the variability of individual entries. By imposing a much weaker finite fourth moment condition compared with the sub-Gaussianity condition in the literature, we derive the optimal rate of convergence under the spectral norm. In addition, we also provide theoretical guarantee on support recovery. The adaptive thresholding procedure of the MOM estimator is easy to implement and gains robustness when outliers or heavy-tailedness exist. Thorough simulation studies are conducted to show the advantages of the proposed procedure over some state-of-the-arts methods. At last, we apply the proposed method to analyze a microbiome dataset in human gut. The R script for implementing the method is available at https://github.com/heyongstat/RCEC.