Well, its kinda like how dogs can be treated with a lot of cancer drugs people can't- its largely a matter of longevity and personhood. Just load 'em up with stuff that'll cause all kinds of crazy side effects in a decade or so, if they're at least six, even in longer lived breeds. (assuming its a bad form of cancer in the first place)

Sheep don't have to worry about the things people have to worry about when they aren't going to have to use the same body for the better part of a century.
So yes, they do get sick, but its not regarded as the same level of devastation because its an animal, you can use harsher treatments, and you're not looking at the same level of magnitude of suffering or life lost.

If we had much shorter lifespans, we'd honestly be some pretty rugged creatures. Our stomachs are only .5 less acidic than some carrion birds. More acidic than most carnivores.
No worries about preserving teeth or joints, eat what you want, its literally Yolo!

Our brains, vision, bipedalism and weird vomiting thing are pretty damn sensitive though.
A lot of animals don't have such a vomit issue. Rats literally can't vomit, iirc.