Answer: C

Video solution:

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Written solution:

Place $A=(0,0,0),$ $B=(2,0,0),$ $C=(0,3,0),$ and $D=(0,0,4).$ Then

$$\begin{aligned} BC &= \sqrt{2^2+3^2}=\sqrt{13}, \\ BD &= \sqrt{2^2+4^2}=2\sqrt5, \\ CD &= \sqrt{3^2+4^2}=5, \end{aligned}$$

so this coordinate model matches all the given edge lengths. The tetrahedron is a rectangular-corner tetrahedron with perpendicular edge lengths $2,3,4$ from $A,$ so its volume is

$$\frac16(2)(3)(4)=4.$$

Thus, C is the correct answer.