Though there is a difference in every individual on how much and till when would this be, for us singles, that amounts to your cost of living for 6 months without work. Someone said 3 months would do but since this is an emergency fund, I might as well do the 6 months mark. This would also be your source for whatever inevitable thing happens-- accidents, medical issues, death of a relative or whatever emergency means to you (puhlease. that cute bag you saw on sale is not an emergency). And when that time comes, no matter how much you spent, you must save back till the last cent to ensure that you have more of it for the next incident.
I'm around 30% of my emergency fund as of this writing and my goal is to be done with it before this year ends. And because this is an amount that I needed to have anytime it is needed, I plan to invest just half of it to RTBs and the other half at my disposal so I can benefit from the interests gained in the tenure of the coupon and still be able to make use of it when something happens (don't want to be disappointed by all the fees that I might get when I withdraw prematurely from the bond). And the interest? Everything would just be invested back in low risk investments once I accumulated enough.
There is also a thing called insurance. But really. An emergency is something urgent. You don't want to be held back by some stupid company who has a gazillion clause so you'd get as little as possible. Have an insurance if you want, but also keep an emergency fund on your own. Not only is it much accessible, it is less paperwork and headache for you.