The problem is affecting iPhone, iPad and Android phone users, according to posts on the Microsoft Answers forum. It also, according to some, is affecting some Windows Phone and Surface RT users, as well. (Thanks to @doylebob for the link to the Answers forum threads.)
The problem seems to stem from issues with Exchange ActiveSync, the Microsoft protocol for accessing Exchange Server natively from mobile devices.
On January 10, I asked Microsoft for an update on the issue, which still seems only partially resolved. A spokesperson emailed me the following statement:
"Microsoft is investigating an issue affecting a small percentage of mobile users' access to Hotmail and Outlook.com, and we are working to restore full access to the service as quickly as possible. For the latest information, we encourage users to visit the Hotmail and Outlook.com status page."
(Note: The status page gives information specific to whichever Microsoft account is signed in.)
A post by a moderator about the ongoing access problems provided this update:
"Hotmail is working for all web users and POP-based accounts. Some mobile device users relying on EAS (Active Sync) may see problems accessing their email, calendar and contacts.
"Please check the status page for updates and ETA on a fix."
While most, if not all, mobile Hotmail and Outlook.com users currently can access their mail via a Web site, some still cannot using their local mail app on their devices if that mail app relies on EAS. Because some of these users are seeing status pagees indicating all is working, they are becoming increasingly frustrated. It seems as though Microsoft is marking the issue as resolved because users can still get to their mail via Web access.
Update: Around 8:45 pm PT/11:45 pm ET on January 10, Microsoft officials said they believed the Hotmail/Outlook.com EAS issue was resolved and that all users on all mobile platforms should be able to access their mail via the Web and EAS connections.