There are many plugins which while not on their own could achieve this, like infinite scroll. The idea would be that while a user was answering a question, rather than just putting up a banner saying to reload the page. instead you could have a different request which grabbed the new answers and plugged them into the page so you can see the new answers as you type.
As the answer section is at the bottom of the page you could have the issue of it disappearing from below the cursor as new answers were added, but this could be easily solved with the scrollTop property.
You would probably still want a notification that new content had been added but it would not need to be so obtrusive, you could use a nice fade-in / fade out effect which requires no interaction to keep the user informed.
You could even further extend this to comments and edits, so the user could see the comments appear to their answer as they appear, so edits can be made promptly. Though this last part is less of an annoyance than having to move your focus from the question you are answering to click a link to update the content.
Edit: OK, there appears to have been a number of comments that seem to imply that I was suggesting a huge interruption to the workflow of the user, I was not. I was instead trying to suggest the process be made more streamlined and not interrupt the user with new posts, i.e. not have a notification requiring action.
Stack-overflow already has an instant notification (well recent changes) with the message that appears at the top of the page. What I was suggesting was to do away with that or not require interaction, just update the posts. I also was not suggesting a chat like functionality, I realize there is a section for that already. What I would say is many web usability books, taking just 1 of the top of my head "Don't Make Me Think!: A Common Sense Approach to Web Usability" by Steve Krug talks about systems should be making the lives of the users easier not harder. It may not seem like much to automate a step, but the benefits of the user not having to take their attention off the thing they are writing to click something, sure you can still notify just don't interrupt.
As to how, this is a huge community of highly skilled individuals. Does anyone think that this is an insurmountable problem?