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.