The mobile version of Duolingo, for Android, just gives away the correct answer on these types of exercises! Bad idea for learning a language.
This construction is unnatural. It would sound natural to say, "Eating in the laboratory is forbidden." You can also say, "It is forbidden to eat in the laboratory."
Есть = to eat.
Eating = ем, ешь, ест, едим, едят. Also a noun, the act of eating, that's why it can be the subject at the beginning of the sentence "Eating is forbidden."
Word order in English sentences is much more strict than in Russian .
It's not quite grammatically correct in English. It should be
In the laboratory, eating is not allowed. Or, eating is not allowed in the laboratory.
Is it gramatically incorrect to say at the laboratory? I mean, it's a big and public place, so I think at should be accepted.
I reported it anyway.
It's correct both ways but "eating in the laboratory" sounds better than "eating in laboratory"
I took есть to mean "to be" here, is that valid or how would you say that?
That would be быть (to be), not есть (is/are). The verb in this case would be находиться (to be located). Нельзя находится в лаборатории.
Is "you cannot eat in the library / one cannot eat in the library" wrong?
"You cannot eat..." is not wrong, but this Russian sentence would be written on a sign in the library. Your translation would not be on a sign. It is conversational. It should probably be accepted. "One cannot eat..." is strange though. It sounds like people have a medical problem with eating. Maybe "one may not eat..." would be better.