THIS ANSWER IS WRONG (and therefore should not have been accepted)
Suppose that $X$ is a $K(\pi,1)$ and a CW complex. Let $E$ be a universal covering space of $X$, so that $E$ is contractible and there is a free $\pi$-action on $E$ with orbit space $E/\pi\cong X$. (I really mean free and properly discontinuous -- what Hatcher's book calls a "covering space action".)
This gives $L(E)$ a $\pi$-action, again free, with orbit space $L(E)/\pi\cong L_0X$. And $L(E)$ has also $S^1$ acting, commuting with the $\pi$-action, in such a way that $$ (L(E)/S^1)/\pi=L(E)/(S^1\times\pi)=(L(E)/\pi)/S^1\cong L_0(X)/S^1. $$ On the other hand, a deformation retraction of $E$ to a point $p$ yields an $S^1$-equivariant deformation retraction of $L(E)$ to $L(p)$ (i.e. one that is compatible with the $S^1$-action), and this in turn yields a deformation retraction of $L(E)/S^1$ to $L(p)/S^1$.
So $L(E)/S^1$ is contractible and has a free action of $\pi$ with orbit space $L_0(X)/S^1$. This implies that $L_0(X)/S^1$ is a $K(\pi,1)$.