It doesn't make sense to keep money under the mattress or in an account that gives you barely 0.5% annual interest, that's true. But an emergency fund is still crucial: if tomorrow your car breaks down or you lose your job, having three or four months of expenses saved keeps you from going into debt at 20% interest. I learned this the hard way when I had to put an unexpected car repair on my credit card because I had nothing set aside.
What really needs to change is where you put that money. If the economy is unstable, an emergency fund in actual euros (even if it slowly loses purchasing power) is better than nothing; for money you actually want to grow, you need to find something that at least keeps pace with inflation, whether that's index funds, bonds, or whatever works for your situation. The problem isn't saving, it's saving badly.