UPDATE: Microsoft's having a little bit of a messaging problem. The company told Polygon that Phil Harrison's comments on connection requirements were just a "potential scenario," and that it has not officially confirmed any related information yet. The original story follows below.
Xbox One needs a regular internet connection to function. That's not the same as a continuous connection as feared, according to a Kotaku interview with Microsoft VP Phil Harrison, but use of the console is still limited.
"[Xbox One] requires, at some point in the beginning and at various times through its on state, to connect to our cloud and to our internet," Harrison said. "That is to deliver Xbox Live functionality, that is to deliver download content to you, that is to deliver some of the innovations around TV and entertainment that we showed today. But it doesn’t require it to be online all the time."
How often it will need a connection depends on the situation. But to start up, it will need to be online. And to keep playing a game--even a single-player game--it will need to be online once every 24 hours, Harrison said.
So assuming your connection goes bad but you leave your Xbox One on, you could keep playing The Elder Scrolls VII: Orc Country for a full day before running into problems.
This brings a bit more clarity to an odd statement on Xbox One's FAQ: "No, it does not have to be always connected, but Xbox One does require a connection to the Internet."
You don't have to remain connected to the internet at all times, but if you want to use your Xbox One, you'll need regular access of some kind.