We present the results from a sensitive multiwavelength analysis of the properties of extremely red objects (EROs). Our analysis employs deepRIzJHK photometry of an 85 × 85 region to select a sample of 68 EROs with (R-K) ≥ 5.3 and brighter thanK = 20.5 (5 σ). We combine this photometric data set with an extremely deep 1.4 GHz radio map of the field obtained from the VLA. This map reaches a 1 σ limiting flux density of 3.5 μJy, making it the deepest 1.4 GHz map taken, and is sensitive enough to detect an active galaxy withL_1.4 ≳ 10²³ W Hz^-1 atz > 1. If powered by a starburst, this radio luminosity is equivalent to a star formation rate of ≳25M_☉ yr^-1 for stars more massive than 5M_☉. We identify radio counterparts to 21 of the EROs in this field with radio fluxes above 12.6 μJy and resolve one-third of these with our 16 FWHM beam. The spectral energy distributions of the majority of these galaxies are consistent with those expected for dust-reddened starbursts atz ~ 1. At these redshifts the radio luminosities of these galaxies indicate a median far-infrared luminosity of this population ofL_FIR ≳ 10¹²L_☉, meaning that half of the radio-detected sample are ultraluminous infrared galaxies (ULIRGs). We conclude that ≳16% ± 5% of the ERO population brighter thanK = 20.5 are luminous infrared galaxies (LIRGs) atz ~ 1. We also use photometric classification of the colors of the EROs to investigate the mix of dusty active and evolved passive systems in the remaining ERO population that is undetected in our radio map. Based on this we suggest that at least 30% and possibly up to ~60% ofall EROs with (R-K) ≥ 5.3 andK ≤ 20.5 are dusty, star-forming systems atz ≳ 1. Our best estimate of the star formation density in this highly obscured and optically faint (R ≳ 26) population is_*(0.1-100M_☉) = 0.11 ± 0.03M_☉ yr^-1 Mpc^-3, comparable to estimates of that in Hα-emitting galaxies atz ~ 1 and greater than the estimates from UV-selected samples at these epochs. This lends support to the claims of a strong increase in the contribution from obscured systems to the star formation density at high redshifts. Using the observed counts of the radio-detected ERO population, we model the apparent break in theK-band number counts of the whole ERO population atK ~ 19-20 and propose that the passive ERO class dominates the total population in a relatively narrow magnitude range aroundK ≲ 20, with dusty, active EROs making up the bulk of the population at fainter limits.