How should I save money if the real interest rate (after inflation) is negative?
Inflation protected securities (i-bonds or TIPS).
TIPS stands for Treasury Inflation Protected Securities. By very definition, they tend to protect your savings against inflation. They won't beat inflation, but will keep up with it. TIPS or iBonds have two parts. A fixed interest part and a variable interest portion which varies depending upon the current rates. The combined rate would match the inflation rate. They can be bought directly from the treasury (or from a broker or bank who might charge a commission)
(Real) interest rates are so low because governments want people to use their money to improve the economy by spending or investing rather than saving. Their idea is that by consuming or investing you will help to create jobs that will employ people who will spend or invest their pay, and so on. If you want to keep this money for the future you don't want to spend it and interest rates make saving unrewarding therefore you ought to invest.
That was the why, now the how. Inflation protected securities, mentioned in another answer, are the least risk way to do this. These are government guaranteed and very unlikely to default. On the other hand deflation will cause bigger problems for you and the returns will be pitiful compared with historical interest rates. So what else can be done? Investing in companies is one way of improving returns but risk starts to increase so you need to decide what risk profile is right for you.
Investing in companies does not mean having to put money into the stock market either directly or indirectly (through funds) although index tracker funds have good returns and low risk. The corporate bond market is lower risk for a lesser reward than the stock market but with better returns than current interest rates. Investment grade bonds are very low risk, especially in the current economic climate and there are exchange traded funds (ETFs) to diversify more risk away.
Since you don't mention willingness to take risk or the kind of amounts that you have to save I've tried to give some low risk options beyond "buy something inflation linked" but you need to take care to understand the risks of any product you buy or use, be they a bank account, TIPS, bond investments or whatever. Avoid anything that you don't fully understand.