The Reshetnyak majorization theorem (see [9.56][1]) states that any closed rectifiable curve $\alpha$ in a CAT(0) length space $U$ can be *majorized* by a convex plane figure $F$; that is, there is *short* (= 1-Lipschitz) map (= majorization) $m\colon F\to U$ such that $m|_{\partial F}$ is the arc-length parametrization of $\alpha$.

Note that majorization does not decrease the curvature; that is, curvature of $\alpha$ cannot be smaller than curvature of $\partial F$ at the corresponding point.
Therefore, the inequality
$$ \ell \ge 2\cdot \pi\cdot \varepsilon^{-1}$$
holds in CAT(0) spaces as well.


  [1]: https://arxiv.org/pdf/1903.08539v5.pdf