The first time around, it was overnight stocking at a grocery store. That job is truly thankless, and stocking 45 cases per hour (company minimum) is physically strenuous. The second time around was moving mailboxes and doing break/fix for Microsoft's BPOS service.
BPOS was backed by Exchange 2007, I'm pretty sure, which required manual mailbox moves - you had to offline the mailbox to move its files, and we were basically manually rebalancing the mailboxes for thousands of users as each server's local storage became full.
Obviously it's very disruptive to offline a mailbox - the user can't interact with it at all, their client usually has a copy of some of their mail, but they can't check for mail, send new mail, save anything, or search for old mail. And all incoming mail is usually queued. So we had to do this during a region's nighttime. This made trouble for our customers who worked in i.e. the USA branch of an European company; they would have to deal with EMEA outage hours, which were something like 2pm to 10pm PST.
We were a 24/7 operation with a 15-minute SLA for emergencies, and we covered the after-hours alerts for several other groups as well. We provided worldwide support for the service. Us mailbox movers worked alongside the emergency oncall break/fix engineers.