Most are aware of the current service outage, and I would encourage you all to have some patience, while the relevant teams work to get things up and running again.
A temporary workaround to use your content, until services are restored, is to set your console to 'Offline' via Settings > Network Settings. By doing this, you should be able to sign in locally, until any outage is resolved. More on the process, here. This will NOT work on games that require an internet connection.
If you don't perform this step, your profile will continue to attempt to connect to Xbox Live, FAIL, and will play a sign in - sign out game, over and over again, causing you MORE frustration.
All Ambassadors, Moderators, the Xbox Support Team and fellow gamers are trying to help, by keeping up with numerous threads, so please have a little consideration for us folks. Creating more, only makes the problem worse.
Keep up to date on the outage, by visiting the Xbox Live Status page.
Hope this little bit of useful info, helps.