Ultraslow radiative cooling lifetimes and adiabatic detachment energies for three astrochemically relevant anions, Cn- (n = 3-5), are measured using the Double ElectroStatic Ion Ring ExpEriment (DESIREE) infrastructure at Stockholm University. DESIREE maintains a background pressure of approximate to 10(-14) mbar and temperature of approximate to 13 K, allowing storage of mass-selected ions for hours and providing conditions coined a "molecular cloud in a box." Here, we construct two-dimensional (2D) photodetachment spectra for the target anions by recording photodetachment signal as a function of irradiation wavelength and ion storage time (seconds to minute time scale). Ion cooling lifetimes, which are associated with infrared radiative emission, are extracted from the 2D photodetachment spectrum for each ion by tracking the disappearance of vibrational hot-band signal with ion storage time, giving 1e cooling lifetimes of 3.1 +/- 0.1 s (C3-), 6.8 +/- 0.5 s (C4-), and 24 +/- 5 s (C5-). Fits of the photodetachment spectra for cold ions, i.e., those stored for at least 30 s, provide adiabatic detachment energies in good agreement with values from laser photoelectron spectroscopy on jet-cooled anions, confirming that radiative cooling has occurred in DESIREE. Ion cooling lifetimes are simulated using a simple harmonic cascade model, finding good agreement with experiment and providing a mode-by-mode understanding of the radiative cooling properties. The 2D photodetachment strategy and radiative cooling modeling developed in this study could be applied to investigate the ultraslow cooling dynamics of a wide range of molecular anions.