The variance reduction class of algorithms including the representative ones, SVRG and SARAH, have well documented merits for empirical risk minimization tasks. However, they require grid search to optimally tune parameters (step size and the number of iterations per inner loop) for best performance. This work introduces `almost tune-free' SVRG and SARAH schemes by equipping them with Barzilai-Borwein (BB) step sizes. Along with the BB step size, both i) averaging schemes; and, ii) the inner loop length that adjusted according to the BB step size are required for the best performance. In particular, SVRG, SARAH and their BB variants are reexamined through an `estimate sequence' lens to enable new averaging methods that tighten their the convergence rates theoretically, and improve their performance empirically when the step size or the inner loop length is chosen large. Then a simple yet effective means to adjust the number of iterations per inner loop is developed, which completes the tune-free variance reduction together with BB step sizes. Numerical tests corroborate the proposed methods.