We study the square-root LASSO method for high-dimensional sparse linear models with weakly dependent errors. The asymptotic and non-asymptotic bounds for the estimation errors are derived. Our results cover a wide range of weakly dependent errors, including -mixing, -mixing, phi-mixing, and m-dependent types. Numerical simulations are conducted to show the consistency property of square-root LASSO. An empirical application to financial data highlights the importance of the results and method.