We present the results from a sensitive multiwavelength analysis of the properties of extremely red objects (EROs). Our analysis employs deep RIzJHK photometry of an 8.'5 x 8.'5 region to select a sample of 68 EROs with (R-K) greater than or equal to 5.3 and brighter than K = 20.5(5sigma). 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 sigma limiting flux density of 3.5 muJy, making it the deepest 1.4 GHz map taken, and is sensitive enough to detect an active galaxy with L-1.4 greater than or similar to 10(23) WHz(-1) at z > 1. If powered by a starburst, this radio luminosity is equivalent to a star formation rate of greater than or similar to 25 M-. yr(-1) for stars more massive than 5 M-.. We identify radio counterparts to 21 of the EROs in this field with radio fluxes above 12.6 muJy and resolve one-third of these with our 1."6 FWHM beam. The spectral energy distributions of the majority of these galaxies are consistent with those expected for dust-reddened starbursts at z similar to 1. At these redshifts the radio luminosities of these galaxies indicate a median far-infrared luminosity of this population of L-FIR greater than or similar to 10(12) L-., meaning that half of the radio-detected sample are ultraluminous infrared galaxies (ULIRGs). We conclude that greater than or similar to16% +/- 5% of the ERO population brighter than K = 20.5 are luminous infrared galaxies (LIRGs) at z similar to 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 similar to60% of all EROs with (R-K) greater than or equal to 5.3 and K less than or equal to 20.5 are dusty, star-forming systems at z greater than or similar to 1. Our best estimate of the star formation density in this highly obscured and optically faint (R greater than or similar to 26) population is (rho) over dot(*) (0.1-100 M-.) = 0.11 +/- 0.03 M-. yr(-1) Mpc(-3), comparable to estimates of that in Halpha-emitting galaxies at z 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 the K-band number counts of the whole ERO population at K 19 20 and propose that the passive ERO class dominates the total population in a relatively narrow magnitude range around K less than or similar to 20, with dusty, active EROs making up the bulk of the population at fainter limits.