The brain performs intelligent tasks with extremely low energy consumption. This work takes its inspiration from two strategies used by the brain to achieve this energy efficiency: the absence of separation between computing and memory functions and reliance on low-precision computation. The emergence of resistive memory technologies indeed provides an opportunity to tightly co-integrate logic and memory in hardware. In parallel, the recently proposed concept of a Binarized Neural Network, where multiplications are replaced by exclusive NOR (XNOR) logic gates, offers a way to implement artificial intelligence using very low precision computation. In this work, we therefore propose a strategy for implementing low-energy Binarized Neural Networks that employs brain-inspired concepts while retaining the energy benefits of digital electronics. We design, fabricate, and test a memory array, including periphery and sensing circuits, that is optimized for this in-memory computing scheme. Our circuit employs hafnium oxide resistive memory integrated in the back end of line of a 130-nm CMOS process, in a two-transistor, two-resistor cell, which allows the exclusive NOR operations of the neural network to be performed directly within the sense amplifiers. We show, based on extensive electrical measurements, that our design allows a reduction in the number of bit errors on the synaptic weights without the use of formal error-correcting codes. We design a whole system using this memory array. We show on standard machine learning tasks (MNIST, CIFAR-10, ImageNet, and an ECG task) that the system has inherent resilience to bit errors. We evidence that its energy consumption is attractive compared to more standard approaches and that it can use memory devices in regimes where they exhibit particularly low programming energy and high endurance. We conclude the work by discussing how it associates biologically plausible ideas with more traditional digital electronics concepts.