Deep learning has seen tremendous interest in medical imaging, particularly in the use of convolutional neural networks (CNNs) for developing automated diagnostic tools. The facility of its non-invasive acquisition makes retinal fundus imaging amenable to such automated approaches. Recent work in analyzing fundus images using CNNs relies on access to massive data for training and validation - hundreds of thousands of images. However, data residency and data privacy restrictions stymie the applicability of this approach in medical settings where patient confidentiality is a mandate. Here, we showcase results for the performance of DL on small datasets to classify patient sex from fundus images - a trait thought not to be present or quantifiable in fundus images until recently. We fine-tune a Resnet-152 model whose last layer has been modified for binary classification. In several experiments, we assess performance in the small dataset context using one private (DOVS) and one public (ODIR) data source. Our models, developed using approximately 2500 fundus images, achieved test AUC scores of up to 0.72 (95% CI: [0.67, 0.77]). This corresponds to a mere 25% decrease in performance despite a nearly 1000-fold decrease in the dataset size compared to prior work in the literature. Even with a hard task like sex categorization from retinal images, we find that classification is possible with very small datasets. Additionally, we perform domain adaptation experiments between DOVS and ODIR; explore the effect of data curation on training and generalizability; and investigate model ensembling to maximize CNN classifier performance in the context of small development datasets.