  We analyze the topological properties of the possible superconducting states
emerging from a Cd$_3$As$_2$-like, $\mathcal{C}_4$-symmetric Dirac semimetal,
with two four-fold degenerate Dirac points separated in the $k_z$ direction.
Unlike the simplest Weyl semimetal for which all pairing orders are
topologically obstructed and nodal, we show that the topological obstruction
for pairing in Dirac semimetals crucially only exists for certain pairing
symmetries. In particular, we focus on odd-parity $B_{1u}$ and $B_{2u}$ pairing
states, both of which can be induced by Ising ferromagnetic fluctuations. The
$B_{1u}$ and $B_{2u}$ pairing states inherit the topological obstruction from
the normal state, which dictates that these states necessarily hosts four
Bogolibov- de Gennes (BdG) Dirac point nodes protected by a $\mathbb{Z}_2$
monopole charge. By a Wannier state analysis, we show that the topological
obstruction in the superconducting states is of higher-order nature. As a
result, in a rod geometry with gapped surfaces, arcs of higher-order Majorana
zero modes exist in certain $k_z$ regions of the hinges between the BdG Dirac
points. Unlike Fermi arcs in Weyl semimetals, the higher-order Majorana arcs
are stable against self-annihilation due to an additional $\mathbb{Z}$-valued
monopole charge of the BdG Dirac points protected by $\mathcal{C}_4$ symmetry.
We find that the same $\mathbb{Z}$-valued charge is also carried by $B_{1g}$
and $B_{2g}$ channels, where the BdG spectrum hosts bulk ``nodal cages", i.e.,
cages formed by nodal lines, that are stable against symmetry preserving
perturbations.
