We present an approach to deep neural network based (DNN-based) distance estimation in reverberant rooms for supporting geometry calibration tasks in wireless acoustic sensor networks. Signal diffuseness information from acoustic signals is aggregated via the coherent-to-diffuse power ratio to obtain a distance-related feature, which is mapped to a source-to-microphone distance estimate by means of a DNN. This information is then combined with direction-of-arrival estimates from compact microphone arrays to infer the geometry of the sensor network. Unlike many other approaches to geometry calibration, the proposed scheme does only require that the sampling clocks of the sensor nodes are roughly synchronized. In simulations we show that the proposed DNN-based distance estimator generalizes to unseen acoustic environments and that precise estimates of the sensor node positions are obtained.