Can I become a professor if I were expelled from the uni?
Yes, you can become a professor. Your expulsion (for unsatisfactory marks) seems equivalent to failing your degree. Failing a degree doesn't preclude becoming professor. But you'll (likely) need a degree to become one. (The definition of professor varies between countries.)
From comments:

[is] failing a degree...dishonourable...something [to] be ashamed of?

Failure isn't shameful nor disgraceful: A professor who hasn't failed, hasn't challenged themselves enough.

Do you think, as a professor, you have the right to keep it to yourself or be transparent?

In the immediate future, universities and employers will want to know why you dropped out, failed, were excluded, however you want to phrase it. Just be honest. Explain your situation. You were burning out, deteriorating mentally and physically, these aren't shameful/disgraceful events.
A professor (depending on definition) probably need never mention distant failures: In the future, historic details (even failing a degree) become irrelevant.