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Performance-wise, how does HDF5 compare to a relational database?

It really depends on your application. HDF5 is tuned to do efficient I/O and storage for "big" data (hundreds of megabytes and more). It will not work well for small reads/writes.

It doesn't have indexing capabilities, though we are working on some limited features.

HDF5 was designed to complement DBs and not to compete with them.