The trouble is, the list of things that could happen is huge, because most 3 year olds have no sense of safety - I spend much of my time saying to my 4 year old "Please don't climb on the back of the chair like that - if you fall, it will hurt." We have had big discussions about why it would hurt, what other things have hurt her and how she felt, she looks at me all wide eyed and says "Oh yes Mummy, I remember" and then does the same thing half an hour later. Same with wearing a seat belt - she still tries to wriggle out of it on occasion. Because she just doesn't get it yet. Things that have hurt her in the past, she gets and avoids (like if I tell her food is hot, she'll wait and blow on it first because she's learnt that lesson!) but if it hasn't entered her experience yet, she can't be trusted to remember what I've told her.
So you've got all the issues with fire and incidents along those lines, with the child not knowing what to do, plus the fact she's vomiting and could hurt herself (bang her head, say, or choke) or just be scared, plus the fact that she could then be unpredictable - she could be climbing furniture to see where daddy is.
The link below is from family who were in the house when their child climbed her chest of drawers, somehow pulled it over and didn't survive. It's upsetting.
www.meghanshope.org/
You are definitely not over-reacting, and she is far too young to be on her own.
I was in the kitchen when my two year old let herself out of our garden (which I didn't think she could do, or would as she'd never shown any interest in the garden gate up to then), and took herself for a walk along the path by the lake opposite our house. Ten minutes of me cooking her dinner before I realised it was quiet outside and stuck my head out. Ten minutes of me going frantic trying to work out which way she might have gone - round the lake, or up to the main road.... - and cursing myself for not having stuck my head out sooner and for not noticing the odd quiet in the garden sooner and then again for just not having checked every minute at least. You just don't know what they can or will do, and you don't want to be at preschool when they try it at home...