The answer depends on what types of emergencies you want to protect against.
The most common protection is for loss of income. Typically this means job loss, and in this case you should think about how easy it would be for you to get another job, and how much it would pay. Certain people are in high demand and could have a new similar paying job in 24 hours, whereas others might take 6 months or longer to find similar paying work. Some people are overpaid and know it would be impossible to find a similar paying job. So, first try to honestly assess how long it would take you to find a new job. Second, calculate your expenses needed to cover you during that time. Third, multiply that number by 2 (in case you're wrong and to avoid stress). Lastly, take the higher of your number or 60 days. (I recommend a minimum of 60 days because some companies like to start new hires on a particular day of the month, and then you still may not got paid for at least 3 weeks after your first day.)
If you are protecting against a medical emergency then you should have at least your insurance deductible available in your emergency fund. If you are unlucky enough to be injured while out of work, you should add your deductible to the amount you calculated for job loss above. Note though, that most medical providers will offer some sort of payment plan if you are unable to pay all at once. Even without a payment plan most large medical claims take months to process before you receive the final bill, and this would give you time to plan accordingly. If your insurance plan offers an HSA, I'd recommend trying to build up your balance to be at least your deductible, and if you can afford it I'd recommend maxing out the contributions even if you don't need it (because you can use it as a tax-advantaged retirement account too).
If you can think of other emergencies, factor them in too. Do you have a relative that may be sick or in need of financial assistance, and you know you'll be helping out? Do you have a crazy terrier mix dog that could jump out the window of a moving car and break his leg which could cost $5K to repair? In most cases you don't have to add the totals of every type of emergency into a large number, but certainly be conscious of the fact that more than one emergency type could happen at the same time.
Once you've calculated how much you want to keep in your emergency fund, I highly recommend opening a Roth IRA if you haven't already, and storing your emergency fund there. Make sure that the bank you choose makes it easy for you to take money out of the account (in case of emergency). The reason the Roth works so well is that if you have an emergency you can take any amount you've ever contributed out at any time without penalty. (But not the investment growth beyond your contributions; if you do that then you do pay a penalty.) Once you've withdrawn the money, you may choose to undo the distribution within 60 days without affecting that year's maximum contribution (currently $5500). The best part is, if you don't have an emergency, you can leave it in the Roth and invest it with tax free growth for life. If you can afford it, I'd strive to max out your Roth IRA each year and let it double as your emergency fund. If you have even more money left over, then you can set that aside as your (new) emergency fund so you don't have to touch the Roth.