"I feel unwell and I need a hospital."
Translation:Me siento mal y necesito un hospital.
For talking about internal feelings (well, unwell, sad, happy, etc.) the reflexive form "sentirse" is usually used.
Me siento mal, ¿Te sientes bien?, Se siente triste.
I believe it's because "feel" in this sentence is acting as a reflexive verb. The action "reflects" the state of the sentence subject and in spanish reflexive sentences require "me" for yo forms, "te" for tú forms or "se" for usted and ustedes forms, "nos" for nosotros forms.
The whole sentence would be written as "Yo me siento mal y necesito un hospital" and could be translated as "I myself feel bad and I need a hospital."