Motivated by the importance ascribed to correlations in random matrices used to model phenomena in various scientific disciplines, we report how algebraic correlations between matrix elements affect the eigenvalue statistics and spectral density of random matrices. These correlations, introduced through thresholded Gaussian fields on a square lattice, decay as a power law $\propto r^{-2H}$, with exponent $H>0$. As $H$ varies, both the eigenvalue distribution and excess kurtosis undergo qualitative changes. The spectral variance approaches $1/2$, while the moment of order $2n$ is finite only for $H>(n-1)/(2n)$. Long-wavelength scaling gives an algebraic density tail $|\lambda {|}^{-1-2/(1-2H)}$ for $0<H<1/2$ and the corresponding finite-size powers and logarithms. At $H=1/2$ the support is unbounded although every fixed moment is finite; for $H>1/2$ it is bounded and approaches the semicircle as $H\to \infty$. A self-consistent equation determines the limiting density from the binary covariance and fixes the excess kurtosis through the squared correlations along a row. Our results link real-space correlations to spectral regimes, distinguishing the roles of the long-distance exponent and the full covariance.