Major Nelson assured us that at the end of this generation's lifecycle, when the Xbox One's servers shut down permanently, you'll still be able to play your old games.
In an interview with /r/games, Larry "Major Nelson" Hryb answered some of the most "upvoted" questions from the popular videogaming sub-reddit. Right off the bat, he was asked what would happen after the Xbox One servers are shut down at the end of the new generation. Specifically, he was asked if games would be unplayable after Microsoft pulled the plug. His response was a reassuring one: "That's certainly something we would not do. That's not the way the system is designed. It's designed for flexibility. But let's get the system out there first."
Next, Hryb was asked whether players who were banned from Xbox Live would lose access to the single player portion of all of the games they have registered to their account. "Absolutely not," Hryb replied with confidence.