If it is for sequences and related data, CHADO database scheme from the GMOD project provides a good start. This is qnot really software, just a common standard. It is best suited for data storage for genome browsers or websites, because of existing scripts and other CHADO-aware software. Needs PostgreSQL for running and Perl for data preparation, loading and reporting. This would be one of the more complex solutions, I suppose.
From my experience, for a biological database you can use the same tools that you would use for any other database. So it all depends on your data. Is it small, big, huge? You can go from flat files to relational or object-oriented or semantic databases. Some other possibilities are working with no-sql approaches as well for instance hadoop or mongodb.