The crystal structure, magnetic ground state, and the temperature-dependent microscopic spin-spin correlations of the frustrated honeycomb lattice antiferromagnet Na₂Co₂TeO₆ have been investigated by powder neutron diffraction. A long-range antiferromagnetic (AFM) ordering has been found below T_N ∼ 24.8K. The magnetic ground state, determined to be zigzag antiferromagnetic and characterized by a propagation vector k = (1/200), occurs due to the competing exchange interactions up to third-nearest neighborswithin the honeycomb lattice. The exceptional existence of a limited magnetic correlation length along the c axis (perpendicular to the honeycomb layers in the ab planes) has been found even at 1.8 K, well belowthe TN ∼ 24.8K. The observed limited correlation along the c axis is explained by the disorder distribution of the Na ions within the intermediate layers between honeycomb planes. The reduced ordered moments m_Co(1) = 2.77(3)μB/Co²⁺ and m_Co(2) = 2.45(2)μB/Co²⁺ at 1.8 K reflect the persistence of spin fluctuations in the ordered state. Above TN ∼ 24.8K, the presence of short-range magnetic correlations, manifested by broad diffuse magnetic peaks in the diffraction patterns, has been found. Reverse Monte Carlo analysis of the experimental diffuse magnetic scattering data reveals that the spin correlations are mainly confined within the two-dimensional honeycomb layers (ab plane) with a correlation length of ∼12 Å at 25 K. The nature of the spin arrangements is found to be similar in both the short-range and long-range ordered magnetic states. This implies that the short-range correlation grows with decreasing temperature and leads to the zigzag AFM ordering at T ≤ TN. The present study provides a comprehensive picture of the magnetic correlations over the temperature range above and below the TN and their relation to the crystal structure. The role of intermediate soft Na layers on the magnetic coupling between honeycomb planes is discussed.