Is it fun taking care of a baby?

I could be wrong, but when you say "up to your elbows in poo", I'm assuming you mean the same thing as when someone says,"I'm up to my ears in housework" or "I'm up to my neck in bills."

The time and energy involved in caring for a baby is not fun or immediately rewarding. Babies have no "off" button -- they are constantly requiring some type of care or attention, and often at the least convenient times and places.

(It's especially upsetting if the baby has colic, because the non-stop glass-shattering screams can lower your immune system, interfere with your digestion, increase your heart rate and blood pressure, give you a headache, prevent sleep, and cause anxiety, irritability, and even rage.)

So the short answer to your question is "no", at least not under the circumstances you've described. But it's okay (and normal) to express anger about this situation. That's what this website is for.

some people think it is

What kind of baby are you taking care of that you have poop on your elbow?  And at around 4 months the baby should be sleeping through most nights