We introduce the LUNA neutron detector array developed for the investigation of the ¹³C(α, n)¹⁶O reaction towards its astrophysical s-process Gamow peak in the low-background environment of the Laboratori Nazionali del Gran Sasso (LNGS). Eighteen ³He counters are arranged in two different configurations (in a vertical and a horizontal orientation) to optimize neutron detection efficiency, target handling and target cooling over the investigated energy range E_α,lab=300−400 keV (E_n=2.2−2.6MeV in emitted neutron energy). As a result of the deep underground location, the passive shielding of the setup and active background suppression using pulse shape discrimination, we reached a total background rate of 1.23±0.12 counts/hour. This resulted in an improvement of two orders of magnitude over the state of the art allowing a direct measurement of the ¹³C(α, n)¹⁶O cross-section down to E_α,lab=300 keV. The absolute neutron detection efficiency of the setup was determined using the ⁵¹V(p,n)⁵¹Cr reaction and an AmBe radioactive source, and completed with a Geant4 simulation. We determined a (34 ± 3)% and (38 ± 3)% detection efficiency for the vertical and horizontal configurations, respectively, for E_n=2.4MeV neutrons.