  This paper studies the action of symplectic homeomorphisms on smooth
submanifolds, with a main focus on the behaviour of symplectic homeomorphisms
with respect to numerical invariants like capacities. Our main result is that a
symplectic homeomorphism may preserve and squeeze codimension $4$ symplectic
submanifolds ($C^0$-flexibility), while this is impossible for codimension $2$
symplectic submanifolds ($C^0$-rigidity). We also discuss $C^0$-invariants of
coistropic and Lagrangian submanifolds, proving some rigidity results and
formulating some conjectures. We finally formulate an Eliashberg-Gromov
$C^0$-rigidity type question for submanifolds, which we solve in many cases.
Our main technical tool is a quantitative $h$-principle result in symplectic
geometry.
