Does this sound correct? “What do you do with the food you cannot finish in a restaurant?” “I would get them to go.””I would take them to go.”

I would get it to go. I would take it to go.

The actual wording of the response will vary regionally but the correct pronoun is ‘it’ rather than ‘them’ (“I’d take it to go” etc). In Australia we would likely say “I’d get a doggy bag” because that’s the slang term for the take away container / bag. Otherwise we might say “I’d take it home” or something like that.

"I would take it to go" "I would get a box for leftovers" "I would take it home"

I agree with the other commenter that “I would get/take it to go” sounds natural. The term “to go” in this context is pretty American (in my experience). It can be used to describe both taking home leftover food from a restaurant, and going to a restaurant to pick up food and leave. In British English, it is more common to order food “take away” instead of “to go”. As another commenter said, in some places (not common in the US) people will ask for a “doggy bag”, to indicate they want to take their leftover food home in a bag.

Sounds fine, except “food” isn’t plural so you’d take “it”, not “them”

Just so you know, in UK and US English this is different. If you told a British waiter you would like your food to go he would be confused instead It's I would like a takeaway box.

Yes

In American, the best way to say it to the wait staff is, "Would you please help me get this boxed up to go?" or "I would like a take out container, please." or "Can I have a box for the leftovers?" Others have described how to tell the story afterward.

"I would take IT to go."

Of those two options…”take them” is better. But we’d normally say “I’d take it home” or “I’d get a [take home] box” 

Tip (US based)…at a restaurant, ask your server “can I get a box, please?” And they’ll either give you a container for you to transfer your leftovers into or will take it back to the kitchen and transfer it (nicer the place the higher the chance they’ll do it for you) Also servers will often ask if there’s still food on the plate when you ask for a check, “Are you all done or would you like a box?” You can reply I’m done, I’m still working on it (means you don’t want to take it home but you want to eat a bit more), or I’d like a box. Servers will ask that even if it doesn’t look like there’s enough food for another meal or you look done…this is because some people really don’t like food waste, and it also lets them know if there was something wrong with the food if there’s a lot of leftovers and there’s not a box.  So if I’m traveling and didn’t finish my meal, I might say something like “I’m done..it’s a shame my hotel room doesn’t have a fridge.”

I would ask for a doggy bag.” Or “I would ask for a to-go box.”

"I'd take it home." "I'd get it wrapped up to take home." "I'd ask them to put it in a to-go/take-out box."

First, I would refer to the food as “it” rather than “them.” If you’re specifically talking about unfinished food in a restaurant, I would use “get it to go.”

No, it sounds strange. You'd just say that you'd leave it. "The pizza was enormous, so I left half of it."