Estimation of the disturbance or interference covariance matrix plays a central role in radar target detection. Traditional maximum likelihood (ML) estimators lead to degraded false alarm and detection performance in the realistic regime of limited training. For this reason, structured covariance estimators have been actively researched. This paper reviews as well as proposes new structured covariance estimation methods which exploit physically motivated practical constraints. We first review the rank constrained maximum likelihood (RCML) estimator which explicitly incorporates the rank of the clutter subspace as a constraint in the ML problem. Next, we introduce an efficient approximation of structured covariance under joint Toeplitz and rank constraint (EASTR). In particular, we propose new quadratic optimization problems that enforce Toeplitz structure while preserving rank. Crucially, both the RCML estimator and the EASTR admit closed form solutions and hence facilitate real time implementation. We perform experimental evaluation in the form of normalized SINR, probability of detection, and whiteness tests. In each case, we compare against widely used existing estimators and show that exploiting the practical constraints has significant merits in covariance estimation.