"I have been working on a project recently" is clearly a perfectly grammatical sentence using the Perfect Continuous tense.
It is not clear whether "I have been being sick recently" is grammatically correct though. Of course, most English speakers would simply say "I have been sick lately", but does this necessarily have the same connotations as the other sentence? Perhaps "been" entails the continuous aspect, but I'm not certain.
Similarly, can I say "Starting tomorrow, I will have been being sick for two months now"?
By the way, these sentences read less awkward if you put the emphasis on the word "being", since this implicitly groups "being sick" together.
Edit: In lieu of Ham and Bacon's answer, one can replace every instance of "sick" with "confused" or some other adjective and the question remains the same.
Best Answer
Actually, "being sick" has the implication of 'throwing up', as in "During the entire boat trip, John was being sick," or "I was being sick all over the table". "I have been sick" means that you have been unwell, or ill, due to a disease.
They're both perfectly alright, they just mean differently.
Edit: In lieu of Rolfer's change of question:
Let's take the adjective "happy". The two examples are now "I have been happy" and "I have been being happy".
The first of the two sentences means that you were happy in times past. You might not necessarily be happy now, but you were happy before.
The second of the sentences however, meant that you have been doing some activity which occurs when you are happy, for example, Dancing, or jumping, or shouting, or singing, etc. "I have been being happy" denotes that you were doing an action associated with happiness. This can also be seen by "sick", or "angry" ('being angry' could mean stamping your foot, shouting, saying bad things, etc.).
Essentially, they mean the same thing, just stating it in a different way.