It's possible to recover enough to cleanly save the user work and shutdown. You have to mesure of much memory you need for a clean shutdown and allocate a byte[sizeForShutdown] at the start of the application in the top level exception handler (the only one that should catch that error) and free that byte array before doing anything else.
Now you can argue that this is not truly recovering from exception but it is a lot better than what you can do with most of the other Error subclasses. I said most because there are easy to recover from Error like the StackOverflowError where you just have to fail the operation or request that caused the error.
Does the JLS actually guarantee that if you free that byte array then the extra memory will be immediately available? I thought there could potentially be some lag. The safer approach would be to create all of the objects you need for a clean shutdown during program launch and keep them around. Then you won't need to allocate any memory in the exception handler.
Now you can argue that this is not truly recovering from exception but it is a lot better than what you can do with most of the other Error subclasses. I said most because there are easy to recover from Error like the StackOverflowError where you just have to fail the operation or request that caused the error.