Bin packing is an algorithmic problem that arises in diverse applications such as remnant inventory systems, shipping logistics, and appointment scheduling. In its simplest variant, a sequence of $T$ items (e.g., orders for raw material, packages for delivery) is revealed one at a time, and each item must be packed on arrival in an available bin (e.g., remnant pieces of raw material in inventory, shipping containers). The sizes of items are i.i.d. samples from an unknown distribution, but the sizes are known when the items arrive. The goal is to minimize the number of non-empty bins (equivalently waste, defined to be the total unused space in non-empty bins). This problem has been extensively studied in the Operations Research and Theoretical Computer Science communities, yet all existing heuristics either rely on learning the distribution or exhibit $o(T)$ additive suboptimality compared to the optimal offline algorithm only for certain classes of distributions (those with sublinear optimal expected waste). In this paper, we propose a family of algorithms which are the first truly distribution-oblivious algorithms for stochastic bin packing, and achieve $\mathcal{O}(\sqrt{T})$ additive suboptimality for all item size distributions. Our algorithms are inspired by approximate interior-point algorithms for convex optimization. In addition to regret guarantees for discrete i.i.d. sequences, we extend our results to continuous item size distribution with bounded density, and also prove a family of novel regret bounds for non-i.i.d. input sequences. To the best of our knowledge these are the first such results for non-i.i.d. and non-random-permutation input sequences for online stochastic packing.