We propose a distributionally robust formulation for simultaneously estimating the covariance matrix and the precision matrix of a random vector.The proposed model minimizes the worst-case weighted sum of the Frobenius loss of the covariance estimator and Stein's loss of the precision matrix estimator against all distributions from an ambiguity set centered at the nominal distribution. The radius of the ambiguity set is measured via convex spectral divergence. We demonstrate that the proposed distributionally robust estimation model can be reduced to a convex optimization problem, thereby yielding quasi-analytical estimators. The joint estimators are shown to be nonlinear shrinkage estimators. The eigenvalues of the estimators are shrunk nonlinearly towards a positive scalar, where the scalar is determined by the weight coefficient of the loss terms. By tuning the coefficient carefully, the shrinkage corrects the spectral bias of the empirical covariance/precision matrix estimator. By this property, we call the proposed joint estimator the Spectral concentrated COvariance and Precision matrix Estimator (SCOPE). We demonstrate that the shrinkage effect improves the condition number of the estimator. We provide a parameter-tuning scheme that adjusts the shrinkage target and intensity that is asymptotically optimal. Numerical experiments on synthetic and real data show that our shrinkage estimators perform competitively against state-of-the-art estimators in practical applications.