Multilevel spin toque transfer RAM (STT-RAM) is a suitable storage device for energy-efficient neural network accelerators (NNAs), which relies on large-capacity on-chip memory to support brain-inspired large-scale learning models from conventional artificial neural networks to current popular deep convolutional neural networks. In this paper, we investigate the application of multilevel STT-RAM to general-purpose NNAs. First, the error-resilience feature of neural networks is leveraged to tolerate the read/write reliability issue in multilevel cell STT-RAM using approximate computing. The induced read/write failures at the expense of higher storage density can be effectively masked by a wide spectrum of NN applications with intrinsic forgiveness. Second, we present a precision-tunable STT-RAM buffer for the popular general-purpose NNA. The targeted STT-RAM memory design is able to transform between multiple working modes and adaptable to meet the varying quality constraint of approximate applications. Lastly, the reconfigurable STT-RAM buffer not only enables precision scaling in NNA but also provides adaptiveness to the demand for different learning models with distinct working-set sizes. Particularly, we demonstrate the concept of capacity/precision-tunable STT-RAM memory with the emerging reconfigurable deep NNA and elaborate on the data mapping and storage mode switching policy in STT-RAM memory to achieve the best energy efficiency of approximate computing.