How small can turbojet engines be made whilst remaining fuel efficient, light, cheap, and producing enough thrust to carry a human, gas, and casing?
The figure of merit here is thrust to weight ratio (https://en.m.wikipedia.org/wiki/Thrust-to-weight_ratio)  For a good turbojet, you get about 5:1. So a 100 lb engine could produce about 500 lbf of thrust. Modern High bypass turbofans are significantly more fuel efficient than turbojets, but the thrust to weight ratio is still about 5:1. So calculate how big your human is,  how much gas you want, and divide by 5