The transport properties of the semimetallic quasi-one-dimensional S=1/2 antiferromagnet Yb4As3 have been studied by performing low-temperature (T≥0.02 K) and high magneticfield (B≤60 T) measurements of the electrical resistivity ρ(T, B). For T ≿ 2 K a ‘heavy-fermion’-like behavior Δρ(T)=AT2 with huge and nearly field-independent coefficient A ≈ 3 µΩ cm/K2 is observed, whereas at lower temperatures ρ(T) deviates from this behavior and slightly increases to the lowest T. In B>0 and T ≾ 6 K the resistivity shows an anomalous magnetic-history dependence together with an unusual relaxation behavior. In the isothermal resistivity Shubnikov-de Haas (SdH) oscillations, arising from a low-density system of mobile As-4p holes, with a frequency of 25 T have been recorded. From the T- and B-dependence of the SdH oscillations an effective carrier mass of (0.275±0.005)m0 and a charge-carrier mean-free path of 215 Å are determined. Furthermore, in B≥15 T, the system is near the quantum limit and spin-splitting effects are observed.