How big should your emergency fund be, well that is wholly dependent on how flexible your spending is, if the emergency actually cuts off your job income OR increases your spending, # of jobs you have, their stability, your health, your living circumstances etc.
Here's a fun guideline, your emergency fund should cover one (risky), two (medium), or 3 (conservative) of the following (assign a dollar value to each and pick the 1/2/3 most expensive ones and add them up):
1) Complete loss of income, 2 months, bills/essential expenses ONLY, PLUS COBRA (no UI). (if you pick 2 or 3, you could instead double or triple this if this one is the most expensive)
2) Combine your driver and home/renters insurance deductible "drive car thru house".
3) Your Out of Pocket Maximum healthcare coverage for you or your family (double if someone is chronically ill)
4) Do you drive? If so, the most expensive possible repair, or the price of a car you'd be willing to buy cash instead of repairing
5) Do you own a home? if so, insert the most expensive home repair here that you are liable for (usually going to be roof, A/C, or Boiler)
6) Do you rent? Three month's rent in case you need to pick up and move (for deposit and moving costs)
7) "death" fund. Regardless of life insurance, money to unexpectedly need to bury a loved one, plus time off from work, counseling, and maybe a month or two of unexpected daycare while you pick up the pieces / wait on the life insurance money.
8) "lawyer" fund. 40 hours at the going billable rate of a lawyer you might find yourself in need of at the going rate in your locale. More if you are especially legally vulnerable.
For me, that is
1) $8000
2) $5000
3) $7000
4) $5000 (I don't drive expensive cars and can get around by bike, so this one barely applies to me)
5) $6000 (A/C, no heat (FL) and HOA covers the roof
6) N/A
7) $4000 funeral, $1000 daycare, $1000 off work = $6000
8) 40*250/hr = $10,000
So my cash-money Efund should be $10,000, I'm perfectly happy only securing one of these, I figure I could secure another $8000 (two-fer) or $15,000 (triple-threat) through other means such as selling something, no/low interest APR CC, or a roth withdrawl.