This is a very interesting thread topic, it was fun to read through the replies!
For me, my emotions flicker between extremes and exhausted apathy. The latter is infinite and deep, like a yawning abyss. I feel disconnected from everything and as if my consciousness is floating in darkness. As for the rest:
Sadness I'd describe as crushing, physically and mentally. It's unbearable. I'd rather apathy, every time. I'm usually too busy spiraling to focus on bodily sensations, but I can at least say that I always have to urge to scream or to start babbling nonsense to myself, so I suppose part of it must be in my throat.
Anger makes my whole body tense, especially my limbs. It can affect my jaw, too. Since I'm focused on suppressing it I tend to be very "in my head", expressing it internally rather than externally, and I barely speak. It usually feels as though I'm about to boil over and lose control due to how much I've bottled up over my lifetime.
When I'm anxious, I tend to make myself as small as I can to try and feel safer. I often hide my face behind my hair, and I hold my arms close to me with a hunched sort of posture.
My happiness is mainly in my shoulders and hands, I think, which is strange. I fidget a lot, and my thoughts feel light. Also, it's rare, but there are times when I've been so overwhelmed by joy that I've had to stop to calm myself down because it turned uncomfortable. I would start to experience very bad sensory overload, and my excitement would turn into uncontrollable physical anxiety, making me shaky and dissociated.
Oh- And if I experience particularly strong emotions of any kind whether positive or negative, my chest gets very tight, and I feel like my heart is in a vice grip.