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 network-accessible block data storage volumes that may be used by programs executing on other physical computing systems. 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. If a program using a volume becomes unavailable, another program (e.g., another copy of the same program) may in some situations obtain access to and continue to use the same volume, such as in an automatic manner in some such situations.