  Neutrinoless double-beta ($0\nu\beta\beta$) decay and the standard
two-neutrino double-beta ($2\nu\beta\beta$) decay of $^{98}$Mo have been
studied. The double-beta decay $Q$-value has been determined as
$Q_{\beta\beta}=113.668(68)$ keV using the JYFLTRAP Penning trap mass
spectrometer. It is in agreement with the literature value,
$Q_{\beta\beta}=109(6)$ keV, but almost 90 times more precise. Based on the
measured $Q$-value, precise phase-space factors for $2\nu\beta\beta$ decay and
$0\nu\beta\beta$ decay, needed in the half-life predictions, have been
calculated. Furthermore, the involved nuclear matrix elements have been
computed in the proton-neutron quasiparticle random-phase approximation
(pnQRPA) and the microscopic interacting boson model (IBM-2) frameworks.
Finally, predictions for the $2\nu\beta\beta$ decay are given, suggesting a
much longer half-life than for the currently observed cases.
