This paper assumes a set of identical wireless hosts, each one aware of its location. The network is described by a unit distance graph whose vertices are points on the plane two of which are connected if their distance is at most one. The goal of this paper is to design local distributed solutions that require a constant number of communication rounds, independently of the network size or diameter. This is achieved through a combination of distributed computing and computational complexity tools. Starting with a unit distance graph, the paper shows: 1. How to extract a triangulated planar spanner; 2. Several algorithms are proposed to construct spanning trees of the triangulation. Also, it is described how to construct three spanning trees of the Delaunay triangulation having pairwise empty intersection, with high probability. These algorithms are interesting in their own right, since trees are a popular structure used by many network algorithms; 3. A load balanced distributed storage strategy on top of the trees is presented, that spreads replicas of data stored in the hosts in a way that the difference between the number of replicas stored by any two hosts is small. Each of the algorithms presented is local, and hence so is the final distributed storage solution, obtained by composing all of them. This implies that the solution adapts very quickly, in constant time, to network topology changes. We present a thorough experimental evaluation of each of the algorithms supporting our claims.