It is true that we don't need a linguistic context to say "I'm tired," but that is because the context is the "here-and-now" of the speech situation. We do, however, need a context to say "I was tired," since the simple past is appropriate only with an established past context, for example, "I was tired by eight o'clock last night, so I went to bed" or "She saw that I was tired, so she stopped peppering me with questions."
The sentence, "I have been tired" does indeed look strange by itself. It is, however, perfectly grammatical. Unlike verbs denoting events, as a stative verb phrase it needs an appropriate context.
With verbs denoting events, no temporal context is needed for the present perfect, e.g. The mail has arrived; Jerome has lost his cat. But a verb phrase denoting a state is different; in order to be used in the present perfect it needs a time frame. Here are a few examples of appropriate contexts:
1) A: You don't look so good. Is there anything the matter?
B: No, it's just that I've been tired lately. I don't know why. (lately= adverb of time)
2) A: You don't look so good. Is there anything the matter?
B: Well, I've been tired for the past two weeks. I guess it's time to see my doctor. (for + time period)
3) A: You look really tired. Is there anything the matter?
B: I'm totally exhausted. I've been tired, but never like this. I guess it's time to see my doctor. (a state at an unspecified time before the present)
Other such examples are
She has needed someone to boss around all her life
Why have you never understood me?
We've known for months that they weren't getting along
Often a grammatical utterance does turn out to be nonsensical. The eminent Danish grammarian Otto Jespersen (1860-1943), in a book discussing language teaching, poked fun at a grammatically impeccable sentence he once found in an English textbook: "Your horse had been old." It would indeed be hard to find a context that would redeem such an utterance.