We consider unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise. We show that while there are infinitely many unbiased estimators for this problem, none of them has uniformly minimum variance. Therefore, we focus on locally minimum variance unbiased (LMVU) estimators. We derive simple closed-form lower and upper bounds on the variance of LMVU estimators or, equivalently, on the Barankin bound (BB). Our bounds allow an estimation of the threshold region separating the low-SNR and high-SNR regimes, and they indicate the asymptotic behavior of the BB at high SNR. We also develop numerical lower and upper bounds which are tighter than the closed-form bounds and thus characterize the BB more accurately. Numerical studies compare our characterization of the BB with established biased estimation schemes, and demonstrate that while unbiased estimators perform poorly at low SNR, they may perform better than biased estimators at high SNR. An interesting conclusion of our analysis is that the high-SNR behavior of the BB depends solely on the value of the smallest nonzero component of the sparse vector, and that this type of dependence is also exhibited by the performance of certain practical estimators.