Learn Farsi: نه دکتر در این بیمارستان هستند.

In Farsi, after any number, the noun always stays in its singular form! So instead of saying "nine doctors" with a plural suffix, you literally say "nine doctor" (نه دکتر).

If دکتر is in the singular form, why does the sentence end with the plural verb هستند (they are)?

Even though the noun stays singular after a number, the verb still matches the real meaning of the sentence when you are talking about people. Since "nine doctors" refers to multiple human beings, Farsi uses the plural verb هستند.

Doesn't the word نه also mean "no"? How do I tell them apart?

Yes, the words for "nine" and "no" are spelled exactly the same in Farsi! You just have to tell them apart by the context of the sentence. Here, نه comes right before a noun, which is a big clue that it is acting as a number.

The English translation says "There are...", but I don't see a Farsi word for "there". Why?

Farsi doesn't use a filler word like "there" to show that something exists. You just state the subject, its location, and end with the "to be" verb. The sentence is literally built as "Nine doctor in this hospital are."