Suppose you have to work the night shift, but you're tired and want an evening with your mates.

I've got a family emergency

MIGHT work, certainly won't work more than once and worse - it runs the risk of you being asked to elaborate and make up something on the fly. Instead try something like this:

-Sigh Heavily- Alright, buckle up because it's a long one. So my little sister (Who's turning 17 in march) has been all mixed up about her adolescent sexuality for a while now, but seems to have a new girlfriend. My mum acts like she's really "accepting", but she's one of those "not in my backyard" closeted boomer sexists, so while she'll cheer on gay pride parades, she's totally not willing to accept so much as the POSSIBILITY of somebody being anything other than CIS in her family. So anyway, family lunch and my sister brings her girlfriend. Things are normal at first because mum thinks she's just a friend, y'know. But as the day progresses, it kinda becomes obvious and mum's getting more and more purse lipped by the second. Dad's more ACTUALLY accepting, but also a self proclaimed peacekeeper, so he's trying to get my sister to "Turn down the PDAs in front of your mother" but that, of course, just makes her more vehement to show off. Mum's now getting increasingly annoyed and starts getting snarky. The pair of them get into a proper fight, while dad's trying to get my sister to calm down, which makes her freak out because (of course) she thinks MUM should calm down. Anyway, my sister's partner, my brother and I are just watching this shitstorm unfold, eventually it escalates and my sister threatens to kill herself in front of everyone, dad calls the police, mum breaks down, whole thing is a fecal-fan-fantasia and long story short, I'm now in the emergency room because my sister has to have a psyche evaluation and apparently they do them here, not at the police station. Dad's at home consoling mum and I'm stuck here until everything's sorted because my brother can't drive. Really sorry but I honestly have no idea when I'll be in, if I can make it I'll let you know, but thought I should give you some notice.

Now about 30 seconds into that essay of a made up story, your boss has either completely checked out, or interrupted you to say "It's fine" but in any case, absolutely nobody is going to bother prying into something so personal, messy and just frustrating to listen to.