The popularity of the films Total Recall (1990, 2012) and Eternal Sunshine of the Spotless Mind (2004) tells us that the idea of memory manipulation can make for an interesting sci-fi premise. It's easy to imagine memories—of traumatic PTSD-inducing events, food aversion events, phobia-causing events, and so on—that we'd rather not hold on to. Can science accommodate us? Sort of.
This cartoon from Head Squeeze goes over the processes of conscious memory encoding, consolidation, and retrieval:
Modern-day neuroscience can't competently interfere with conscious memory; these processes are much too refined, and our understanding of them much too clumsy, for that. But Joseph LeDoux, director of New York University's Center for Neuroscience, has discovered a possible workaround: because memory retrieval alters the content of memories themselves, blocking protein synthesis of traumatic memories while they are being recalled can dull emotional responses to them to the point where it becomes possible to forget, or at least ignore, their content. We can already do this for traumatized rats, and since rats (like humans) have a limbic system that mediates emotional responses to memory, there's good reason to think this approach may also work on humans. Some researchers have even suggested that supervised doses of propranolol (inderal), a beta blocker commonly used to treat heart disease and performance anxiety, can be used to help people achieve enough emotional distance from traumatic memories to forget them entirely. (As someone who takes a fairly large dose of extended-release propranolol every morning to prevent migraines, I'm not aware of this feature—though I suppose it's possible I might have been aware of it at some point…)
If we want to completely remove more complex chunks of conscious memory—bad relationships, bad jobs, and so forth—we're out of luck, at least for the time being. But if we want to reduce the emotional power that specific bad memories have over our lives, neuroscientists now have a pretty good general idea of how this can be done.