We study the geometry of the AGN obscurer in IRAS 09104+4109, an IR-luminous, radio-intermediate FR-I source at $z=0.442$, using infrared data from Spitzer and Herschel, X-ray data from Nustar, Swift, Suzaku, and Chandra, and an optical spectrum from Palomar. The infrared data imply a total rest-frame 1-1000$mu$m luminosity of $5.5times10^{46}$erg s$^{-1}$ and require both an AGN torus and starburst model. The AGN torus has an anisotropy-corrected IR luminosity of $4.9times10^{46}$erg s$^{-1}$, and a viewing angle and half opening angle both of approximately $36$ degrees from pole-on. The starburst has a star formation rate of $(110pm34)$M$_{odot}$ yr$^{-1}$ and an age of $<50$Myr. These results are consistent with two epochs of luminous activity in 09104: one approximately $150$Myr ago, and one ongoing. The X-ray data suggest a photon index of $Gamma simeq 1.8$ and a line-of-sight column of $N_{rm H} simeq 5times10^{23}$cm$^{-2}$. This argues against a reflection-dominated hard X-ray spectrum, which would have implied a much higher $N_{rm H}$ and luminosity. The X-ray and infrared data are consistent with a bolometric AGN luminosity of $L_{rm bol}sim(0.5-2.5)times10^{47}$erg s$^{-1}$. The X-ray and infrared data are further consistent with coaligned AGN obscurers in which the line of sight skims the torus. This is also consistent with the optical spectra, which show both coronal iron lines and broad lines in polarized but not direct light. Combining constraints from the X-ray, optical, and infrared data suggests that the AGN obscurer is within a vertical height of $20$pc, and a radius of $125$pc, of the nucleus.
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