We present physical properties of spectroscopically confirmed Lyα emitters (LAEs) with very large rest-frame Lyα equivalent widths EW₀(Lyα). Although the definition of large EW₀(Lyα) LAEs is usually difficult due to limited statistical and systematic uncertainties, we identify six LAEs selected from ∼3000 LAEs at z ∼ 2 with reliable measurements of EW₀ (Lyα) ≃ 200-400 Å given by careful continuum determinations with our deep photometric and spectroscopic data. These large EW₀(Lyα) LAEs do not have signatures of AGN, but notably small stellar masses of M_★ = 10^7-8 M_⊙ and high specific star formation rates (star formation rate per unit galaxy stellar mass) of ∼100 Gyr^-1. These LAEs are characterized by the median values of L(Lyα) = 3.7 × 10⁴² erg s^-1 and M_UV = -18.0 as well as the blue UV continuum slope of β = -2.5 ± 0.2 and the low dust extinction E(B-V)_* = 0.02^{+0.04}_{-0.02}, which indicate a high median Lyα escape fraction of f_esc^{Lyα }=0.68± 0.30. This large f_esc^{Lyα } value is explained by the low H I column density in the interstellar medium which is consistent with full width at half-maximum (FWHM) of the Lyα line, FWHM(Lyα) = 212 ± 32 km s^-1, significantly narrower than those of small EW₀(Lyα) LAEs. Based on the stellar evolution models, our observational constraints of the large EW₀ (Lyα), the small β, and the rest-frame He II EW imply that at least a half of our large EW₀(Lyα) LAEs would have young stellar ages of ≲20 Myr and very low metallicities of Z < 0.02 Z_⊙ regardless of the star formation history.