When Zabbix database is down, any api request correctly fails with a json error message (before 2.2, the error was not json, but just HTML, it was fixed in 2.2.x).
However the error message misses the "data" property:

{"jsonrpc":"2.0","error":{"code":1,"message":"Error connecting to database [No such file or directory]"},"id":0}

while the API documentation ( https://www.zabbix.com/documentation/2.4/manual/api) states that:

"If an error occurred, instead of the result property, the response object will contain an error property with the following data:

• code - an error code;
• message - a short error summary;
• data - a more detailed error message."

so either the documentation needs to be updated or the error in this case, should also contain a "data" property.