Mariano's comment has essentially answered the question, but I'll go ahead and flesh it out.
On any manifold $X$, there is an isomorphism of groups $$\mathrm{Pic}(X) \xrightarrow{~\cong~} H^2(X; \mathbb{Z}),$$ $$L \mapsto c_1(L).$$ Now if $$c_1(L) \equiv 0 \pmod 2,$$ then there is some element $a \in H^2(X; \mathbb{Z})$ such that $$c_1(L) = 2a.$$ The above isomorphism tells us that there exists a complex line bundle $K \in \mathrm{Pic}(X)$ such that $c_1(K) = a$ and $$K \otimes K \mapsto 2a = c_1(L).$$ Then $$K \otimes K \cong L,$$ so that $K$ is a square root of $L$.
For a proof of the above isomorphism, see for example Proposition 3.10 in Allen Hatcher's unfinished book Vector Bundles and $K$-Theory.