We report on the synthesis, crystal structure, magnetic, thermodynamic, and electron-spin-resonance properties of the coordination complex [Cu2(pz)3(4-HOpy)4](ClO4)4 (pz=pyrazine; 4-HOpy=4-hydroxypyridine). This material is identified as a spin-1/2 Heisenberg ladder system with exchange-coupling parameters Jrung/kB=12.1(1)K and Jleg/kB=10.5(3)K [Jrung/Jleg=1.15(4)]. For single crystals our measurements revealed two critical fields, μ0Hc1=4.63(5)T and μ0Hc2=22.78(5)T (for H∥a∗), separating the gapped spin-liquid, gapless Tomonaga-Luttinger-liquid, and fully spin-polarized phase. No signature of a field-induced transition into a magnetically ordered phase was found at temperatures down to 450 mK. The material bridges an important gap by providing an excellent physical realization of an almost isotropic spin-1/2 strong-rung Heisenberg ladder system with modest exchange-coupling energy and critical-field scales.