We investigate the magnetism of the Co^II₄(OH)₂(C₁₀H₁₆O₄)₃ metal-organic framework, which displays complex inorganic chains separated from each other by distances of 1 to 2 nm and orders at 5.4 K. The zero-field magnetic structure is determined using neutron powder diffraction: it is mainly antiferromagnetic but possesses a ferromagnetic component along the c axis. This magnetic structure persists in presence of a magnetic field.Alternating current susceptibility measurements confirm the existence of a single thermally activated regime over seven decades in frequency (E/k_B ≈ 64 K), whereas time-dependent relaxation of the magnetization after saturation in an external field leads to a two times smaller energy barrier. These experiments probe the slow dynamics of domain walls within the chains: we propose that the ac measurements are sensitive to the motion of existing domain walls within the chains, while the magnetization measurements are governed by the creation of domain walls.