We consider a general conic mixed-binary set where each homogeneous conic constraint involves an affine function of independent continuous variables and an epigraph variable associated with a nonnegative function, $f_j$, of common binary variables. Sets of this form naturally arise as substructures in a number of applications including mean-risk optimization, chance-constrained problems, portfolio optimization, lot-sizing and scheduling, fractional programming, variants of the best subset selection problem, and distributionally robust chance-constrained programs. When all of the functions $f_j$'s are submodular, we give a convex hull description of this set that relies on characterizing the epigraphs of $f_j$'s. Our result unifies and generalizes an existing result in two important directions. First, it considers multiple general convex cone constraints instead of a single second-order cone type constraint. Second, it takes arbitrary nonnegative functions instead of a specific submodular function obtained from the square root of an affine function. We close by demonstrating the applicability of our results in the context of a number of broad problem classes.