Cellular microscopy images contain rich insights about biology. To extract this information, researchers use features, or measurements of the patterns of interest in the images. Here, we introduce a convolutional neural network (CNN) to automatically design features for fluorescence microscopy. We use a self-supervised method to learn feature representations of single cells in microscopy images without labelled training data. We train CNNs on a simple task that leverages the inherent structure of microscopy images and controls for variation in cell morphology and imaging: given one cell from an image, the CNN is asked to predict the fluorescence pattern in a second different cell from the same image. We show that our method learns high-quality features that describe protein expression patterns in single cells both yeast and human microscopy datasets. Moreover, we demonstrate that our features are useful for exploratory biological analysis, by capturing high-resolution cellular components in a proteome-wide cluster analysis of human proteins, and by quantifying multi-localized proteins and single-cell variability. We believe paired cell inpainting is a generalizable method to obtain feature representations of single cells in multichannel microscopy images.