Techniques are described for managing access of executing programs to non-local block data storage. In some situations, a block data storage service uses multiple server storage systems to reliably store block data that may be accessed over one or more networks by programs executing on other physical computing systems. Users may create block data storage volumes that are each stored by at least two of the server block data storage systems, and may initiate use of such volumes by one or more executing programs, such as in a reliable manner by enabling an automatic switch to a second volume copy if a first volume copy becomes unavailable. A group of multiple server block data storage systems that store block data volumes may in some situations be co-located at a data center, and programs that use volumes stored there may execute on other physical computing systems at that data center.