Depending on the bulk modulus of the material being stretched and the strength of the interaction with the walls of the chamber holding the material, you may be able to get to several negative atmospheres of pressure. Dark energy also creates negative pressure in otherwise empty space.
As a counter force, the expanding hot gases create a large amount of pressure pushing outward toward the Sun’s surface. The push and pull between gravity and pressure create conditions that maintain the three interior regions of the Sun: the core, the radiative zone, and the convective zone.
Dark energy may be described as a fluid with negative pressure. We say that this negative pressure counteracts gravity and accelerates the expansion of the Universe. Now consider, for example, a star. Gravity contracts the star, but positive (thermal) pressure counteracts the collapse.
If more solar energy comes in, then Earth warms and will emit more heat to space to restore the balance.* Not all of the Sun’s energy that enters Earth’s atmo-sphere makes it to the surface. The atmosphere reflects some of the incoming solar energy back to space immediately and absorbs still more energy before it can reach the surface.
The example you gave, negative "pressure" in a solid is such an example: Engineers quantify the maximum of it that a material can take as ultimate tensile strength. However, pressure does not really describe the situation for solids very well, because forces acting at a surface need not necessarily be normal to that surface.
A tension can't be negative, it can be less then 1 but it can't be less than zero. Try to take the air out of a bottle. Let be the normal air pressure 1unit, you never will be able to reach less then zero. A negative pressure makes no sense