The two-dimensional classical Hardy spaces $H_p\left(ℝ\times ℝ\right)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p\left(ℝ\times ℝ\right)$ to $L_p\left({ℝ}^2\right)$ (1/2 < p ≤ ∞) and is of weak type $(H_1^{}\left(ℝ\times ℝ\right),L_1\left({ℝ}^2\right))$ where the Hardy space $H_1\left(ℝ\times ℝ\right)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^{(}ℝ\times ℝ)$ ⊃ $LlogL\left({ℝ}^2\right)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p\left(ℝ\times ℝ\right)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p\left(ℝ\times ℝ\right)$, the Fejér means...