Set-valued standard Young tableaux (Young tableaux in which the cells are filled with nonempty sets of positive integers) are a generalization of standard Young tableaux due to Buch (2002) with applications in algebraic geometry. The enumeration of set-valued SYT is significantly more complicated than in the ordinary case, although product formulas are known in certain special cases. In this work, we study the case of two-rowed set-valued SYT with a fixed number of entries. These tableaux are a new combinatorial model for the Catalan, Narayana, and Kreweras numbers and can be shown to be in correspondence with both 321-avoiding permutations and a certain class of bicolored Motzkin paths. We also introduce a generalization of the set-valued comajor index studied by Hopkins, Lazar, and Linusson (2023) and use this statistic to find seemingly new q-analogs of the Catalan and Narayana numbers.