Which is correct?

.....and her family is unable to pay for it by themselves.
or
.....and her family is unable to pay for it by itself.

The second sentence is correct. We assume that in this case that the family is one unit.

but you will also often hear

and her family are unable to pay for it by themselves.

where the family is treated as a group of people. (family are treated?)

:-)

The correct sentence would be ".....and her family is unable to pay for it by themselves."

To determine which pronoun to use in this sentence, we need to consider whether the subject, "her family," is singular or plural. In this case, "family" refers to multiple people, so it is plural.

When referring to a plural subject, we use the plural pronoun "themselves." This reflects that the action of paying for something is done collectively by the family members.

On the other hand, if the subject were singular, we would use the singular pronoun "itself." For example, if the sentence were ".....and her family is unable to pay for it on its own," meaning that the family is acting collectively as a unit rather than as individual members.