Update: It looks like the sign-in issues with Xbox Live have been fixed, but Microsoft is now dealing with reports that some users cannot purchase content or downloadable items, or unlock Xbox Live achievements.
The current problem has lasted for about an hour so far. In a post on Microsoft's Xbox Live support page, it says:
"Xbox members, are you having trouble connecting to Xbox Live? We are currently working with our partner to find a fix for these issues. Thanks for being patient. Another update will be provided as soon as possible."
In addition, it appears some users have had issues with accessing the Xbox.com website. Microsoft again says it is "working with our partner to find a solution for this issue."
Thanks to Jeremy for the tip!