Eu sunt bărbat means that I have all the general qualities of a man (biologically, psychologically or both).
Eu sunt un bărbat means that ”I am a man” and it can have more interpretations: I am a man like many others, I am a man (from a specific group of men) etc. In some contexts it might be understood like the first sentence.
So, from en->ro: ”I am a man” can mean both ”Eu sunt bărbat/Eu sunt un bărbat.”
In Romanian, there are general statements that gives a general qualification to the subject like: Eu sunt elev. - I am a pupil. Ea este profesoară. - She is a teacher/professor.