I can not currently visit user profile pages on any site I've tested. Activity pages work just fine.
Visiting anyone else's profile link fails† since links default to the profile tab unless you're looking at your own page or you're on a site you moderate but if you force the page URL to the activity tab, you can visit their activity page with no issue, so it's only the profile part of the page that's broken. For example:
Shog9's default link (this should time out):
Shog9's activity tab link (this should work):
This is currently happening again, same time of day, same day of the week, so I'm guessing there's something running that's causing the profile pages to not populate.
So, something running on Sunday between 4:00 and 5:00 Zulu time.
†: Oddly, some user pages work just fine... so it's not all of them. Found a few staff profile pages that loaded just fine but some non-staff pages also work.