Is "I'm being angry" ungrammatical?

Solution 1:

Yes, “am being ADJECTIVE” sounds quite odd to a native speaker. It is not impossible, but it is very uncommon and seldom what you want.

You would virtually never say “I am being ready” or “I am being happy” instead of “I am ready” or “I am happy”.

If you really, really want a progressive aspect, then it works better with verbs like getting or becoming, depending on the adjective involved and whether these is already an existing idiom.

• I am becoming happy.
• I’m getting happy/happier.
• I’m getting ready.
• I’m getting tired.

Sometimes an adverb is enough:

• I am still happy.

Solution 2:

I wouldn't be so sure that anger is a "temporary state on its own". Some people seem to be angry all the time!

I am being [adjective] can be used to mean that you are only displaying a certain behaviour for a limited period of time.

Contrary to @tchrist, that construction does not strike me as odd at all. In fact, it can convey nuances that the present tense simply cannot. Examples:

He's being facetious; don't pay him any attention.

Here, it's implied that his facetious behaviour, however common, will eventually come to an end. Compare that to:

He's facetious; don't pay him any attention.

The present tense here means that facetiousness is in his nature: in other words, he's always facetious.

I know I'm being overly curious here, but could you be more specific?

(meaning: I'm not usually that curious, but I find that what you said requires elaboration). Compare that to:

I know I'm curious, but could you be more specific?

(meaning: excuse me for always being curious, but I'd like to hear more about that.)

I'll concede that "I am being angry" may sound a bit odd in some contexts. However, I find the following perfectly acceptable:

While you were being angry at the kid for breaking the vase, I attempted to glue the pieces back together.

(meaning: while you were telling the kid off, ...)