Not in any furry closet.
I'm pretty canonical when it comes to the defiition of a furry:
"Someone who appreciates fictional anthropomorphic characters." That's it.
So, to be "a closeted furry" makes as little sense in my mind as being
a closeted Whovian, or a closeted Steampunk.
Furry is a hobby about creativity: we draw, we write stories and create colorful costumes.
That's nothing you should keep in the closet.
There's a lot of overlap in between fan/subcultures, and sometimes you fall into more than one:
I have my own magic/juggling/balloon-twisting act, where I perform as a steampunk/fox character.
To some members in the audience, I have a steampunk character (who happens to be a fox)
others notice the animal character first (oh, and he's a steampunk too.)
The same thing goes for fursuiting: I LARP as a badger (which makes me a beastfolk LARP'er.)
That character is a protagonist from my own fantasy novel (suddenly I'm a writer in character). I attend EuroFurence wearing the exact same fursuit, sword, armor and cape (and presto! I'm a furry.)
Same guy, same costume.
I know there's a subgroup of furries who can't get enough of the adult stuff. I don't share their interest and I don't feel represented by them. So, to me "furry" is all about creativity, and the only furry thing in the closet is a fursuit - and that's just for keeping the dust away.
I should probably add, that I keep "furry" and "therian" as two separate things. I'm not "out" about the latter.