Five metres, about 15 feet, is the general minimum. It is a floor, not a target, and treating it as a target is the mistake most first-timers make.
Plenty of situations need much more. Some sites specify at least 15 metres from elephant seals. Guides routinely ask for 15 to 25 metres or more from territorial animals, and fur seals in particular. Nesting albatrosses can need a bigger buffer than the general rule.
The rule that actually matters is not a number at all. If an animal changes its behaviour because of your group, your group is already too close. Raised head, vocalising, an open mouth, moving toward you, changing direction, or simply stopping resting all count.
A short list of things that are never allowed. Touching, feeding, following, blocking or posing with an animal. Picking up or moving stones near a penguin colony, or offering one to a bird. Standing in a penguin highway, the worn path they use between the colony and the sea, because a bird on that path is usually carrying food for a chick. Putting yourself between any animal and the water.
If a penguin walks up to you, and they do, that is allowed. It is allowed to close the gap. You are not. Your guide will call it, and usually the call is to back off when you safely can.
Bird flu has tightened things further. Some sites now require you to stay standing and keep bags and equipment off the ground. Expect the rules on your trip to be stricter than the ones on a video from five years ago.