I think you could say "I'm 21 years old".

Yes, it's fine. You can stop reading here! It's fine. It is also fine to say "I'm twenty-one years old" or simply "I'm twenty-one." To some extent, if you are answering a question you would take your cue from the question. "What is your age?" "My age is twenty-one." or "I'm twenty-one." (less formal), or "I'm twenty-one years of age." (very formal). "How old are you?" "I'm twenty-one." or, "I'm twenty-one years old." or, even just "Twenty-one."