I'm going to be honest, Ellington really isn't a town where you can be carless. It's pretty rural, not an easy place to walk around, public transit is nonexistent and things you need will not be walking distance from anywhere. You're going to have a hard time finding any short term rental situation there at all, let alone near that clinic. You're should plan for needing to take Ubers around, but Ellington is not a place I would recommend for someone who doesn't drive.