Analyzing human vasculature and vessel-like, tubular structures, such as airways, is crucial for disease diagnosis and treatment. Current methods often rely on small sub-regions or simplified tree-like structures, rendering analysis of entire organ-level networks at clinical resolution computationally challenging. To this end, we propose VAEsselSparse, an efficient encoder-decoder model to obtain a meaningful yet compact representation of the entire organ-level vascular network at sub-millimeter resolution. VAEsselSparse leverages the inherent sparsity of 3D vascular structures via sparse convolutions and attention mechanisms, achieving substantial spatial compression rates of 8 x 8 x 8. We demonstrate superior reconstruction performance compared to dense counterparts and previous methods. Importantly, the resulting latent space retains clinically relevant discriminative features readily usable for classification tasks, such as aneurysm/stenosis or subvariants of the circle of Willis. Moreover, the compact latent space of VAEsselSparse serves as an effective representation for learning vessel-specific priors through generative models, enabling the synthesis of realistic vasculature.