What makes particles stable? Quantum Field Theory says that particles are merely a disturbance in a field.  Virtual particles can pop in and out of existance.  I am made of particles.  The particles that make me must be stable.  What makes particles stable versus temporary virtual particles and quantum tunneling particles?
 A: Short answer: virtual particles are a side-effect of the mathematical formalism of quantum field theory; they can be used to understand observable real particles, which aren't going anywhere.
"Real" particles satisfy an on-shell energy-momentum relation such as $p^\mu p_\mu =m^2$ if we work in units with $c=1$. If you also set $\hbar=1$, we can alternatively describe this as a dispersion relation in terms of the four-frequency $k^\mu:=\hbar^{-1}p^\mu$, viz. $k^\mu k_\mu = m^2$. In field terms, such results follow from the field's Euler-Lagrange equation, which is why they're on-shell results.
"Virtual" particles, which have the short-lived existence you describe, are "off-shell" i.e. don't obey this energy-momentum relation. Mathematically, they're discovered by writing the action $S$ in terms of a source current $J$. The path integral formalism replaces the classical stationary action principle with probability amplitudes proportional to $e^{iS}$. If we Fourier-transform to work in terms of four-frequencies, virtual particles contribute to the physics in inverse proportion to the extent of the energy-momentum relation violation. For example, for a scalar field $\phi$ we have the functional integral $$\int \mathcal{D}\phi e^{iS}\propto e^{iW},\,W:=-\frac{1}{2}\int\frac{d^4 k}{(2\pi)^4}\frac{J(-k) J(k)}{k^2-m^2}.$$
Particle accelerators can transfer four-momentum so as to bring an off-shell particle on-shell, allowing us to detect previously undetectable virtual particles that are now real.
