Here's what we do, which I'm sure isn't suitable for everyone's situation, but it's some food for thought at least:
For convenience and easy access we have one third of our emergency funds (2 months' living expenses) in a combination of bank accounts and physical cash. (Cash is cheap considering current interest rates and it's available even during a power outage.) We have no debt so we could substitute credit cards for this portion of our EF but we've chosen peace of mind: cash and bank accounts are less likely to be canceled or blocked due to fraud etc.
The remaining two thirds is in cheap balanced funds of moderate allocation (~ 65% stock) in taxable accounts. However, I doubled the value of these funds because I expect they might lose up to 50% of their value due to market fluctuations. This allocation is split between two reputable financial providers, just in case one has temporary difficulties (IT infrastructure hacked, for example) and we'd be delayed accessing our emergency funds at the worst moment.
The rest of our investments (taxable and tax-advantaged) is in a complicated and aggressive portfolio that I expect will outperform the mentioned balanced funds by a small to moderate amount in the very long term. By increasing our EF allocation to compensate for the expected 50% drawdown, we forgo the extra gain of the aggressive portfolio on that amount, but it still seems better than using very conservative, low yield investments in the current environment. If interest rates of conservative options (CDs, short term bonds etc.) rise significantly, we'll rethink our EF policy.
With an even higher safety margin (because the maximum likely drawdown of our aggressive portfolio is higher than 50%) it would be possible to fold the investment part of our EF into our regular investments. However, the simplicity of the two balanced funds gives peace of mind: If I'm incapacitated during an emergency, my wife, who's less experienced in managing an investment portfolio, won't have the extra stress of figuring out how much money to withdraw from which of the many funds.
Long story short: I think it's a good idea to find a way to use some of your investments as part of your emergency fund because risky investments are the only thing that provides a reasonable return these days.
I wouldn't say that an EF is unneeded: it really depends on how much and what sort of risk you're comfortable with. We don't mind having a very aggressive, well diversified long term portfolio as the risk can be relatively well understood, but we want to reduce our exposure to non-financial risks, like infrastructure etc. If you save like crazy, as has been suggested and I second that recommendation, your EF will soon be tiny compared to your investments, therefore you can easily afford to have an EF and lose just a little bit on returns.