We report the synthesis, crystal structure and magnetic properties of a new heteronuclear polymeric complex based on non-Kramers Tb ions and carboxylic α-fur = C4H3OCOO ligands: {[Tb2Ba(α-fur)8(H2O)4]·2H2O}n. The α-furoate ligands consolidate 1D zig-zag chains running along the c-axis, formed by Tb2 dimers separated by Ba ions. Ab initio calculations, in combination with the fit of experimental data, predict that the single-ion magnetic ground state is highly anisotropic () and consists of a quasi-doublet with a ΔTb/kB = 3.22 K gap, well separated from the next excited state, while the gap for the Tb2 dimer is Δ2Tb/kB = 2.58 K. Static magnetization and heat capacity measurements show that, magnetically, the system can be modeled as dimers of non-Kramers Tb ions, coupled by an antiferromagnetic intradimer interaction J'*/kB = -1.6 K. Dipolar interactions couple the Tb ions in the dimer with their first neighbour ions along the chain, with J''*/kB = -0.15 K, and with the surrounding ions out of the chain, with maximum J'''*/kB = -0.03 K. Ac susceptibility measurements in H = 0 performed down to 50 mK temperatures have enabled us to observe slow relaxation of magnetization, with an Orbach-like activation energy of U/kB = 1.1 K. It is assigned to the sluggish response of the 3D spin system due to a short-range ordering, possibly enhanced by the presence of disorder caused by defects in the polymeric chains. Under the application of a magnetic field, the system slowly relaxes by two distinct direct processes, strongly affected by a phonon bottleneck effect. We discuss the different relaxational phenomenology of the new complex in comparison with that of the isostructural {[Dy2Ba(α-fur)8(H2O)4]·2H2O}n, differing only in the Kramers nature of the ions, and the mononuclear {Ln(α-fur)3(H2O)3}n (Ln = Tb, Dy) complexes, previously reported.