Why are real and/or nominal interest rates positive?

Only nominal interest rates are supposed to be positive, the real interest rates can be negative.

The fact that nominal interest rates cannot go negative is called the "zero lower bound". First, the interest rates we are talking about are interest rates on risk-less nominal bonds. These instruments are part of the "fixed income securities" since they provide predetermined payments at fixed maturities. So why can't these interest rates be negative?

The argument put forth by Black (1995) relates to no-arbitrage. Put simply, in developed countries, holding cash is always a possible alternative to buying bonds. The nominal interest rate of cash is zero. So, if interest rates on bonds were negative, nobody would be interested in buying bonds, and the price would adjust such that the equilibrium interest rates on bonds are exactly zero. This gives no-arbitrage between holding cash, and buying bonds. The same reasoning works for every financial instrument that provides predetermined interest rates (bank account for instance).

Real interest rates however, are defined by nominal interest rates minus the realized inflation rate. Since the inflation rate can go highly positive, real interest rates are not bounded below by zero.