Let $Y$ be a compact hyperbolic surface. There are only finitely many closed geodesics in $Y$ whose lengths are less that $\ell$. Using the fact that $\pi_1(Y)$ is residually finite (and being a little careful with basepoints) you may then find a finite covering space $X$ of $Y$ with no closed geodesics of length less than $\ell$.

Let $Y$ be a compact hyperbolic surface. There are only finitely many closed geodesics in $Y$ whose lengths are less that $\ell$. Since $\pi_1(Y)$ is residually finite, there is a normal subgroup of finite index in $\pi_1(Y)$ that does not contain anything in the conjugacy classes corresponding to these geodesics. The corresponding finite covering space $X$ of $Y$ has no closed geodesics of length less than $\ell$.