"There is nobody at the hospital today."

It is also acceptable but there will be difference at the emphasis. In Turkish, emphasis of the sentence will be at the word that comes just before the verb. For example, in first one "hiç kimse (nobody)" is the emphasis, but at your sentence it is at "hastanede (at the hospital)" and also yours will be an inverted sentence. P.S: Also emphasis can be understood from the voice that rises at speaking.

It accepted my answer without hiç. As I remember, hiç is used to emphasize