This talk is about the structure of Riemannian 3-manifolds satisfying a lower bound on their scalar curvature. These manifolds are toy models for spatial geometry in general relativity. Our motivational question will be "How flat is an isolated gravitational system with very little total mass?" Objects like gravity wells and black holes can distort geometry without accumulating much mass, making this a subtle question. In addition to discussing progress, I will present a "drawstring" construction, which modifies a manifold near a given curve, reducing its length with negligible damage to a scalar curvature lower bound. Unexpected examples are produced with relevance to a few problems. This construction extends ideas of Basilio-Dodziuk-Sormani and Lee-Naber-Neumayer, and is based on joint work with Kai Xu.