Strong Halpha Emitters (HAEs) dominate the z~4 Lyman-break galaxy population. We have identified local analogs of these HAEs using the Sloan Digital Sky Survey (SDSS). At z<0.4, only 0.04% of galaxies are classified as HAEs with Halpha equivalent widths (>500A) comparable to that of z~4 HAEs. Local HAEs have lower stellar mass and lower ultraviolet (UV) luminosity than z~4 HAEs, yet the Halpha-to-UV luminosity ratio as well as their specific star-formation rate is consistent with that of z~4 HAEs indicating that they are scaled down versions of high-z star-forming galaxies. Compared to the previously studied local analogs of z~2 Lyman break galaxies selected using rest-frame UV, local HAEs show similar UV luminosity surface density, weaker Dn(4000) breaks, lower metallicity and lower stellar mass. This supports the idea that local HAEs are less evolved galaxies than the traditional Lyman break analogs. We are not able to constrain if the star-formation history in local HAEs is powered by mergers or by cosmological cold flow accretion. However, in the stacked spectrum, local HAEs show a strong HeII4686 emission line suggesting a population of young (<10Myr), hot, massive stars similar to that seen in some Wolf-Rayet galaxies. Low [NII]/[OIII] line flux ratios imply that local HAEs are inconsistent with being systems that host bright AGN. Instead, it is highly likely that local HAEs are galaxies with an elevated ionization parameter, either due to a high electron density or large escape fraction of hydrogen ionizing photons as in the case for Wolf-Rayet galaxies.