In Java, you can have many threads. A thread is doing its stuff until it is blocked somewhere (typically, on a mutex or a I/O operation). Of course, this does not block other threads.
The fundamental scenario of multithreaded applications is that you use multiple threads when waiting for a blocked thread would introduce too much waiting. Definition of "too much" here depends entirely on you, but in general, this is how you achieve beter performance through better utilization of resources.
There are some limitations in how threads in Java work, however. Most, if not all of them are when the thread is blocked somewhere "outside" of Java such as in OS call or external (native) library. Theoretically, if native code blocks a thread, Java can not do anything about it. Normally, this should not be a problem unless the native code has a bug.
So in the case of a blocking JDBC response, you would create a new thread which would do other work while first thread is waiting for database to complete. Alternatively, you could make a thread just for doing JDBC. You could make it exactly like you want (with listeners etc.) except for limitations imposed by OS. So it's possible, but it's probably not provided out-of-the-box by JDBC drivers. There is a lot of infrastructure already in core Java which you might find useful (thread pools, workers, synchronized collections). But as with any multithreading, you need to be very careful with accessing data from different threads simultaneously.
Since Java 7, there is also support for non-blocking I/O (NIO). This is almost exactly what you are describing. I/O is offloaded to OS, so your operations return immediately and you get a callback when the operation is finished. However, not all libraries support NIO. For my work, I have never had a reason to use it, because I could always implement the same stuff with my threads at least as good.