How does the value of money affect ones payments?

We all know the story, in Germany during the depression, a woman left a basket full of money outside of a store, and she returned to find the money there, but the basket was stolen.

So if my rent is $1000 a month, and the dollar is now worth 1 cent - what do I pay my landlord, or my mortgage company or my car loan?

Will they still accept what is essentially 1000 pieces of useless paper with George Washington's face on it as monthly payment?