Coincidently, I thought about this a few weeks ago (without any conclusion). I think that I can prove that a "weakly locally separated" algebraic space X with an étale cover Spec(k)->X is of the form Spec(k') if X lives over a field k_0 such that k/*k*_0 is algebraic. If X is not locally separated, this condition does not always hold (take A^1/Z where Z acts by translation and restrict this action to the generic point).
Let K be the algebraic closure of k. Let R=Spec(K) x_ X Spec(K). By assumption
j : R -> Spec(K) x_{k_0} Spec(K)
is an immersion and it is enough to show that this is a closed immersion (since fpqc morphisms descend closed immersions).
We can replace k_0 with its perfect closure. This follows from the observation that R is reduced.
Now, the right-hand scheme is a group scheme over Spec(K). Indeed, it is the fundamental group scheme \pi_0(k_0). It is totally disconnected and all its residue fields are K and the group of K-points is the pro-finite group Gal(K/*k*_0).
R is also totally disconnected and all its points have residue field K. The map
j(K) : R(*K*) -> Gal(K/*k*_0)
is injective and locally closed. Since R(*K*) => K(*K*) is an equivalence relation, it follows that j(K) identifies R(*K*) with a subgroup of Gal(K/*k*_0).
Lemma: A locally closed subgroup H of a topological group G is closed.
pf: The closure of H is a subgroup so we can assume that H is open. It is then easily seen that the complement of H is open.
Thus, R(*K*) is a closed subgroup of Gal(K/*k*_0). In particular, j is a closed immersion.
Remark: If K/*k*_0 is not algebraic, then (if K is algebraically closed) we still have a group structure on the K-points of the fiber product of K over k_0. R(*K*) will be a closed subgroup of this group but it is not clear whether this implies that j is closed.
