Sredinicki means the **exact** $\phi^3$ theory. If you understand the exact solutions to the exact "harmonic oscillator stuffed with $x^3$ term", then you can get a gist of a field theory where the **occupation numbers** replace continuous $x$ of harminic oscillator problem.

If you do not understand the exact solutions to the exact "harmonic oscillator stuffed with $x^3$ term", then you may consider an infinite reflecting wall $U(x)= 0,\; x<0,\; U(x)=+\infty,\; x\ge0$ as an example where the solutions are not localized. (There is a "ground state" with $E=0$, though.)

A "cubic oscillator" with big $g$ is similar to a slightly inclined wall (still reflecting), but with no "bottom" for negative-valued $x$, so there is no minimal $E_0$ for such a potential, no ground state, no localization.