More specifically, are there any databases that don't require secondary storage (e.g. HDD) to provide durability?
Note:This is a follow up of my earlier question.
If you want persistence of transations writing to persistent storage is only real option (you perhaps do not want to build many clusters with independent power supplies in independent data centers and still pray that they never fail simultaneously). On the other hand it depends on how valuable your data is. If it is dispensable then pure in-memory DB with sufficient replication may be appropriate. BTW even HDD may fail after you stored your data on it so here is no ideal solution. You may look at http://www.julianbrowne.com/article/viewer/brewers-cap-theorem to choose replication tradeoffs.
Prevayler http://prevayler.org/ is an example of in-memory system backed up with persistent storage (and the code is extremely simple BTW). Durability is provided via transaction logs that are persisted on appropriate device (e.g. HDD or SSD). Each transaction that modifies data is written into log and the log is used to restore DB state after power failure or database/system restart. Aside from Prevayler I have seen similar scheme used to persist message queues. This is indeed similar to how "classic" RDBMS works except that logs are only data written to underlying storage. The logs can be used for replication also so you may send one copy of log to a live replica other one to HDD. Various combinations are possible of course.