Starter refused to dissengage, battery dead.

[Radon]

I was about to leave Canadian Tire (Auto Parts/Repairs) after picking up some tools, and my starter kicked in and wouldn't stop, even without the keys. I tried desparetely to disconnect the battery, but..

1) Didn't have the right tools
2) The corrosion was bad
3) It was *******' cold outside.

So after running inside asking for a hand, and returning... it had stopped. I'm assuming the battery is dead (It seemed to die too quickly though)...

Why did this happen, and what possibly needs replacing.

Car is an '87. I should also mention that I have a remote starter, although it hasn't been used in quite some time, and was in the off position.

Thanks.