No, not that Google Code is shutting down, per se. The alternatives are better and for active projects there is plenty of time to migrate.
What I object to is that the site will only be preserved in read-only mode for 5 months, despite its popularity and the resulting large number of deep links into it on the Web. Why not forever? git, hg, and I believe svn can all provide read-only access (with some bandwidth overhead) by publishing static files over HTTP. With the addition of a static export of the Google Code pages (mostly doable by crawling; some pagination issues would need to be worked out), Google could host the site going forward on a simple HTTP server without needing to maintain its server-side codebase. And Google is lacking in neither HTTP servers or bandwidth... Belated abuse complaints would remain a problem, but it's easy enough to delete things on request.
Google is obviously not legally obligated to keep the site up, and under current mores I suppose it's not socially obligated either. But I think that when the owning company continues to be highly financially successful and maintenance is easy enough, there should be a social obligation for some kind of archival.
Or at least redirect the subdomain to whatever Archive Team comes up with...