Fire away.

That's pretty much what it means, and it sounds silly to me too.

I think what it means here is that she can no longer be hurt, so go ahead and try.

Fire away actually means to attempt to shoot someone, it does not necessarily mean you can shoot me, it means you can try. So, for example, in the song "Hit me with your best shot" by Pat Benatar, she sings "Hit me with your best shot, fire away!" The implication of the expression is that you won't hit your target.

It most usually means to go ahead and ask a question, or begin a statement, or attempt to do something.

It's a figurative meaning for the literal meaning, just as "I'm bulletproof" is figurative. It can, and likely does mean in this situation, "verbal shooting".

Thank you for your helpful answer, Setanta.

I see. So the singer is implying she's ready to take any verbal abuse from someone, probably the one she loved. Verbal shooting. That's interesting.
Thank you JTT.

fire away

used for giving someone permission to ask a question
"Can I ask you a question?" – "Fire away."

http://www.macmillandictionary.com/dictionary/american/fire-away
==================
fire away

: to speak without hesitation —usually used as an imperative

http://www.merriam-webster.com/dictionary/fire+away

IN THIS SONG, "Fire away" can also mean "bring it on"

There are all kinds of metaphors going on in this song, but the main one is that she tells her lover that she is stronger than titanium, no matter what he shoots her with (says or does to her)

LYRICS
You shout it out,
But I can't hear a word you say
I'm talking loud not saying much
I'm criticized but all your bullets ricochet
You shoot me down, but I get up

[Chorus:]
I'm bulletproof, nothing to lose
Fire away, fire away
Ricochet, you take your aim
Fire away, fire away
You shoot me down but I won't fall
I am titanium
You shoot me down but I won't fall
I am titanium

[Sia:]
Cut me down
But it's you who'll have further to fall
Ghost town and haunted love
Raise your voice, sticks and stones may break my bones
I'm talking loud not saying much

[Chorus:]
I'm bulletproof, nothing to lose
Fire away, fire away
Ricochet, you take your aim
Fire away, fire away
You shoot me down but I won't fall
I am titanium
You shoot me down but I won't fall
I am titanium
I am titanium
I am titanium

[Sia:]
Stone-hard, machine gun
Firing at the ones who run
Stone-hard as bulletproof glass

[Chorus:]
You shoot me down but I won't fall
I am titanium
You shoot me down but I won't fall
I am titanium
You shoot me down but I won't fall
I am titanium
You shoot me down but I won't fall
I am titanium
I am titanium

I didn't expect to get to see the whole lyrics.
It's so nice of you. Thank you so much Punkey.