The ARK's have survived for 10 Billion years since their creation. They are made of Anti-entropic materials and will survive the heat-death of the universe. Though they do wear down, we can't destroy them with any known methods reasonably. We can only shut them down.
And even if you destroyed one, there are thousands.
(For reference, ARK 74 is currently drifting outside the galaxy, slightly bent, after passing within a million miles of the event horizon of the black hole.)