I've seen the nets off of the field in the back (soccer/football/lacrosse?) field at the Mays YMCA on Blanco outside of 1604. Might be available during the day or offseason from kids' sports. Would probably need to be a member.
My brother played club lacrosse in college. When he'd be home during summers, he would chalk up the wall at the local elementary school and use it as target practice with an "automatic" ball return.