If something is unsustainable, there's just no way to keep doing it at the same rate or in the same way. You've been able to keep your new puppy from whining by petting her all night long, but that's unsustainable.
When you can bear something, or continue it, it's sustainable, from the Latin root sustinere, "bear or endure." Unsustainable things can't go on, like unsustainable economic growth that's bound to level out or drop eventually, or an unsustainable expectation that you will always have a perfect grade point average. When it comes to the environment, practices that cause long-term harm or deplete resources are also unsustainable: "Ending unsustainable fishing practices successfully revived the salmon population."