The odds of someone submitting a massive amount of code seems likely on this type of site. Whether or not this is permitted is the first hurdle. If it is permitted, then offloading the code to a pastebin might be question number two. If that is suggested, then should there be an official pastebin? Perhaps we could have one created specifically for stackexchange?
I think it's better to include the code in the post if possible. Some people will not follow links to look at the code. Also, moving the code to a different site makes it harder for people to find the questions from search engines.
I can think of two potential downfalls to including the code in the actual question:
1. Lots of scrolling required for long pieces of code
If this is an issue the first step should always be to ask the OP to remove code not relevant to the question, like what has already been done here. This will make it easier for potential answerers to understand the issue, and will also improve search engine accuracy.
2. Line numbers
Currently there's no option to display line numbers in code snippets. I think we should revive this discussion on meta.SO to see if the SOIS team will consider adding this feature before we jump to an external code hosting site.
I'd say pastebins should be avoided. Some of us can't get to them due to firewalls at work, and it really ruins the continuity of the site. I'd expect everything to be here, so we aren't dependent upon some third party resource going away.
I think the usage of a Pastebin should be discouraged. The problem is that they typically expire (most are 1, 7, or 30 days IIRC). At which point the post on this board become basically useless for future reference.
I think large-ish amounts of code is pretty essential for this site to work. But already there's a question posted with a fairly large chunk of code that's difficult to digest because of all the scrolling required to see it. That's definitely a problem. My reactions are: First, a guideline will have to be established for how much code is too much code; second, there needs a better way to display large-ish amounts of code so that its easier to read. I have no experience with pastebins (never even heard of them until reading this thread) so I have no idea if they would help or not.
Plenty exist now: http://en.wikipedia.org/wiki/Comparison_of_pastebins, though I've not taken the time to actually look through any of these.