DH and I just re-assessed this situation recently. Before, our Efund was for mortgage, utilities (necessary), food and health care. All other things were stopped and we have about 6 months of funds for those four things. But, the more we thought about it (and with a baby on the way) we decided we would need at least one month of child care budgeted into our Efund. If one of us is without a job, we can stop child care and he or she can stay home with the 2 youngest and search for work relying on family to watch kiddos during job interviews, etc. But we just built in a month of child care because if it is not a job loss but some other medical or personal emergency, we may need to continue child care (ie,) if one of us is seriously injured and in the hospital for weeks, the other would want to be with the other as needed and we may need to pay for someone to watch our children if and when family cannot.
Our decision to budget this child care expense came from the following: A good friend has a premature baby in the hospital and is in a difficult position of wanting to be with her baby yet not having child care for her 2 year old son. Her family has helped as much as possible but her DH had to go back to work (he owns his own business, no FMLA and has to work to keep health insurance), her parents both work (and have taken time off) and her inlaws live 3 hours away. She has relied on friends, sister in law, neighbors, etc. She is thinking about having her oldest go to a in-home daycare 2-3 days a week so she can be at the hospital (which is an hour and a half drive from her home each way). Because of the hospital regulations, she cannot be in the ICU with her 2 year old so taking him with her to the hospital each day isn't practical.
So things change and expenses change, too. So now we have about 5 months of Efund but feel this is a good fit for our situation (DH tenured at work, I have a pretty stable govt job). But nothing is guaranteed, especially health issues and the possibility of a bad accident has to be kept in mind, as well as the economy.