You could say either "life" or "lives," but I take them to have different meanings.
The ranch workers were destined to live a horrible life, for they were haunted by loneliness.
In this sentence, I would assume that the loneliness the ranch workers were haunted by is a result of their being ranch workers. Because they are all doing the same thing and lonely for the same reason, they have a horrible life and not lives because there is little separating the individual ranch workers. Metaphorically, they are all living the same life.
The ranch workers were destined to live horrible lives, for they were haunted by loneliness.
I would interpret this to mean that each of the ranch workers was lonely, but that they were not necessarily lonely for the same reason (i.e. because they are ranch workers). Unlike in the above sentence, they are not (metaphorically) living the same life.
I didn't find it particularly easy to articulate that, so if it could be clearer please let me know.
I think context plays a large part in interpreting these sorts of sentences, and even then some sentences may sound nonsensical.
All the people in the audience raised their hands.
We all know that hand-raising usually involves raising only one hand, and we can safely assume that everyone in the audience has at least one hand, so it's clear to the reader that the audience members are individually raising a hand.
In other situations, it may be best to rewrite the sentence. I'm shamelessly stealing this example from EL&U:
Corporations may not have a conscience, but they do have PR departments.
Corporations may not have consciences, but they do have PR departments.
Corporations may not have a conscience, but they do have a PR department.
Which of these is correct? Definitely not the third one (which states that all corporations have a single lack of conscience and a single PR department), and while the the second one is more favourable than the first, these sentences could be rewritten into an unambiguous form:
A corporation may not have a conscience, but it does have a PR department.
Here we run into some other difficulty. Your sentence isn't as simple as this one, mostly because you're using the definite article. You could say:
The ranch workers were each destined to live a horrible life, for they were haunted by loneliness.
By using each, the sentence now refers to the ranch workers individually and separately, so using the singular is unambiguous.
Some examples for you:
Humans adopted many pets in history; for example the cat, the dog, etc.
Humanity adopted many pets in history; for example cats, dogs, etc.
In these examples both "cats" and "the cat" represent the group of historical cats who were adopted, while "Humans" and "Humanity" represent the group of humans who adopted them.
Humans (have) adopted the cat throughout history.
Humanity has adopted the feline throughout history.
Humans have adopted cats throughout history.
Humanity has adopted many a cat throughout history.
In these examples "the cat" and "the feline" and "cats" all represents cats adopted during history, while "Humans" and "Humanity" represent the many people during history that adopted cats. The usage of "many a" in the last example, although not common usage, is an acceptable way to allow me to imply "cat" in that sentence is plural; it implies "many cats".
Best Answer
This is correct because cats is a collective noun, meaning you a referring to a group of something. In this case, a group of cats. This works because the word they also refers to a group of things, such as cats and should be used when describing a group, instead of it or those.