It must have a depth though, there's no possible way it could exist.
If it were possible it's volume would be (depth x area) which as you're saying would be (0 x area) therefore always resulting in 0.
but quantum physics do what they want, and don't necessary follow silly laws like volume=depth*area. Events on quantum scale do not naturally roll into our macro scale due to quantum dilatation, but scientists are able to force events into our measurable realm. Who knows what else is possible when we are able to control that area better.
A century ago massless particles were an absurd idea too.
Perhaps it turns out gravity is a 2D plane into our 3D dimension. We experience its force but it's caused by nothing we can measure. The 2D object redeemer describes might translate a bit differently in the real world than the concept. It is probably not as straightforward as line->square->cube. Dimensions might work in a different way. It is just a simple way to illustrate how different dimensions are constructed. A 2D particle in the 3D world might thus translate as a quantum particle (2D) affecting macro particles (such as protons and neutrons, 3D).
