It's a complicated something else that can have many answers. I'll sketch a few. First let's leave GR and QM out of it.
In classical physics the requirement is that the energy must be bounded from below for stability reasons. If the energy is not bounded from below one can always extract energy by reconfiguring the system in a lower and lower energy state. Then, since only differences in energy have physical significance, one is free to add a constant and redefine the energy scale so that the energy density in some reference state is zero. So one is free to define the zero of potential energy of a baseball at the surface of the Earth as zero or as zero if it is at an infinite separation.
In QM the vacuum state is taken to have zero energy by convention. There is a long history of hemming and hawing here starting with Dirac's negative energy states for his equation for the electron and continuing on through the Casimir effect. In the Casimir effect you can produce a highly localized negative energy density space between two parallel conducting plates. Essentially the parallel plates cut off modes of vibration of the vacuum EM field because of the boundary conditions at the plates. With fewer modes the energy is lowered with respect to the vacuum.
It's when we hit GR that we get the positivity requirement (with exceptions allowed locally like for the Casimir effect). There are several lines of reasoning.
1) The stress-energy-momentum tensor must have positive eigenvalues for physical matter and fields. There is a long Wikipedia article that explains this well with an example of perfect fluids. It is located at
The result is more general than perfect fluids which leads us to....
2) Hawking's chronology protection conjecture. Exotic matter (negative density) implies closed timeline curves and non causal physics. https://journals.aps.org/prd/abstract/1 ... evD.46.603