We performed a high resolution study of ${0}^{+}$ states in $^{134}\mathrm{Ba}$ using the $^{136}\mathrm{Ba}(p,t)$ two-neutron transfer reaction. Our experiment shows a significant portion of the $L=0$ pair-transfer strength concentrated at excited ${0}^{+}$ levels in $^{134}\mathrm{Ba}$. Potential implications in the context of $^{136}\mathrm{Xe}\phantom{\rule{4pt}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{4pt}{0ex}}^{136}\mathrm{Ba}$ neutrinoless double beta decay matrix element calculations are briefly discussed.