This is the question I had meant to ask when I asked this question.: Is there a concise characterization of finite dimensional subspaces of $L^\infty?$ (that's what the discussion with @fedja was really about...)
That's much easier and more standard than finite-dimensional subspaces of $L^1$. The answer is all norms in finite dimensions, or in the unit ball picture, all centrally symmetric convex bodies. Every polytope is a slice of an $n$-cube, so clearly you get all of those. But then the Banach-Alaoglu theorem in this case lets you take limits; the space of $d$-dimensional subspaces of $L^\infty$ is compact. The same argument works for $\ell^\infty$, which embeds in $L^\infty$.
In fact in the case of $\ell^\infty$ you can simply explicitly make every centrally symmetric convex body in $d$ dimensions as an intersection of countably many antipodal pairs of half spaces, and take the corresponding embedding in $\ell^\infty$.