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Fully convolutional U-shaped neural networks have largely been the dominant approach for pixel-wise image segmentation. In this work, we tackle two defects that hinder their deployment in real-world applications: 1) Predictions lack uncertainty quantification that may be crucial to many decision-making systems; 2) Large memory storage and computational consumption demanding extensive hardware resources. To address these issues and improve their practicality we demonstrate a few-parameter compact Bayesian convolutional architecture, that achieves a marginal improvement in accuracy in comparison to related work using significantly fewer parameters and compute operations. The architecture combines parameter-efficient operations such as separable convolutions, bilinear interpolation, multi-scale feature propagation and Bayesian inference for per-pixel uncertainty quantification through Monte Carlo Dropout. The best performing configurations required fewer than 2.5 million parameters on diverse challenging datasets with few observations.