Maybe it sounds crazy, but we can use both the opposite words есть or нет here, if translate the english sentence literally:
The latter would have a flavor of "Do you have an old bed by any chance?" :-)
Why is it acceptable to have none of them?
Because the existence of the bed is implied by asking whether it is old.
I think it didn't accept my answer with есть... should it be an acceptable answer or redundant and unnatural?
Why no 'есть' here?