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 software module of the block data storage service that executes on the first computing system to manage access of the first computing system to the internal networks, configuring a logical local storage device of the first computing system to represent a non-local first block data storage volume provided by the block data storage service, the configuring of the logical local storage device being performed by the executing first node manager software module of the block data storage service and including storing first information indicating that a primary copy of the first block data storage volume is located on a second computing system of the plurality of computing systems, and further including storing second information indicating that a mirror copy of the first block data storage volume is located on a third computing system of the plurality of computing systems, the second and third computing systems each being part of the first group of computing systems; after the configuring of the logical local storage device, receiving one or more data access requests initiated by the first executing program to the logical local storage device of the first computing system, the receiving of the one or more data access requests being performed by the executing first node manager software module of the block data storage service; automatically responding to the received data access requests by retrieving the stored first information and interacting over the internal networks with the primary copy on the second computing system on behalf of the first executing program to perform the received data access requests, the interacting over the internal networks being performed in a manner that is transparent to the first executing program, the automatic responding being performed by the executing first node manager software module of the block data storage service without the executing first node manager software module requesting any indication of a location of the primary copy after having received the one or more data access requests and based on use of the stored first information; 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, the automatic determining being performed by the executing first node manager software module of the block data storage service; 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 retrieving the stored second information and interacting over the internal networks with the mirror copy on the third computing system on behalf of the first executing program to perform the other data access requests, the automatic responding to the received other data access requests being performed by the executing first node manager software module of the block data storage service without the executing first node manager software module requesting any indication of a location of the mirror copy after having received the one or more other data access requests and based on the stored second information.