I figured that login system failure is the issue, rather than the service itself, so I went to google.com and logged in there using the Sign-in link at the top right of the page. Then I clicked the Mail link at the top left. Hey presto -- I could access my Gmail, although it was still slow and cranky (IM contacts wouldn't load, for example). But at least I could access my messages.
I'm not sure if this is just a Mac issue or whether it affects all platforms -- a quick search reveals that it does indeed seem to be surprisingly prevalent. But I thought I'd submit it anyway.

