  We analyze charging-energy-protected Majorana-based qubits, focusing on the
residual dephasing that is present when the distance between Majorana zero
modes (MZMs) is insufficient for full topological protection. We argue that the
leading source of dephasing is $1/f$ charge noise. This noise affects the qubit
as a result of the hybridization energy and charge distribution associated with
weakly-overlapping MZMs, which we calculate using a charge-conserving
formalism. We estimate the coherence time to be hundreds of nanoseconds for
Majorana-based qubits whose MZM separation is $L\sim 5\xi$ (with $\xi$ being
the coherence length). The coherence time grows exponentially with MZM
separation and eventually becomes temperature-limited for $L/\xi \sim 30$.
