1. A method for a computing system of a block data storage service to provide remote block data storage capabilities to executing programs, the method comprising: receiving a request to initiate access of a first executing program to non-local block data storage provided by a block data storage service, the first program executing on a first of a plurality of computing systems that are co-located at a geographical location and that share one or more internal networks, the block data storage service using a first group of multiple other of the plurality of computing systems as block data storage systems that provide block data storage to multiple executing programs, the first computing system not being part of the first group; and under control of a first node manager module of the block data storage service that manages access of the first computing system to the internal networks, receiving one or more data access requests initiated by the first executing program to a logical local storage device of the first computing system that represents the non-local block data storage provided by the block data storage service; automatically responding to the received data access requests by interacting over the internal networks with a second computing system on behalf of the first executing program to perform the received data access requests on a first block data storage volume that has a primary copy stored on the second computing system and that has a mirror copy stored on a third computing system, the second and third computing systems each being part of the first group of block data storage systems, the interacting over the internal networks being performed in a manner that is transparent to the first executing program; after the performing of the received data access requests on the first block data storage volume, automatically determining that the primary copy of the first block data storage volume on the second computing system is unavailable; and after receiving one or more other data access requests initiated by the first executing program to the logical local block storage device of the first computing system, automatically responding to the received other data access requests by interacting over the internal networks with the third computing system on behalf of the first executing program to perform the other data access requests on the first block data storage volume mirror copy on the third computing system.