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We present a measurement of the intrinsic space density of intermediate redshift ($z\sim0.5$), massive ($M_{*} \sim 10^{11} \ \text{M}_{\odot}$), compact ($R_{e} \sim 100$ pc) starburst ($Σ_{SFR} \sim 1000 \ \text{M}_{\odot} \ \text{yr}^{-1} \text{kpc}^{-1}$) galaxies with tidal features indicative of them having undergone recent major mergers. A subset of them host kiloparsec scale, $>1000 \ \text{km}\ \text{s}^{-1}$ outflows and have little indication of AGN activity, suggesting that extreme star formation can be a primary driver of large-scale feedback. The aim for this paper is to calculate their space density so we can place them in a better cosmological context. We do this by empirically modeling the stellar populations of massive, compact starburst galaxies. We determine the average timescale for which galaxies that have recently undergone an extreme nuclear starburst would be targeted and included in our spectroscopically selected sample. We find that massive, compact starburst galaxies targeted by our criteria would be selectable for $\sim 148 ^{+27}_{-24}$ Myr and have an intrinsic space density $n_{\text{CS}} \sim (1.1^{+0.5}_{-0.3}) \times 10^{-6} \ \ \text{Mpc}^{-3}$. This space density is broadly consistent with our $z\sim0.5$ compact starbursts being the most extremely compact and star forming low redshift analogs of the compact star forming galaxies in the early Universe as well as them being the progenitors to a fraction of intermediate redshift post starburst and compact quiescent galaxies.