We study the geometry of the active galactic nucleus (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 μm luminosity of 5.5 × 10$\mathrm{<mrow\; data-mjx-texclass="ORD">46</mrow>}$ erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ and require both an AGN torus and a starburst model. The AGN torus has an anisotropy-corrected IR luminosity of 4.9 × 10$\mathrm{<mrow\; data-mjx-texclass="ORD">46</mrow>}$ erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ and a viewing angle and half-opening angle both of approximately 36° from pole-on. The starburst has a star formation rate of (110 ± 34) $M\odot$ yr$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ and an age of <50 Myr. These results are consistent with two epochs of luminous activity in IRAS 09104+4109: one approximately 150 Myr ago, and one ongoing. The X-ray data suggest a photon index of Γ $\simeq$ 1.8 and a line-of-sight column density of $NH$ $\simeq$ 5 × 10$\mathrm{<mrow\; data-mjx-texclass="ORD">23</mrow>}$ cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-2</mrow>}$. This argues against a reflection-dominated hard X-ray spectrum, which would have implied a much higher $NH$ and luminosity. The X-ray and infrared data are consistent with a bolometric AGN luminosity of $Lbol$ ~ (0.5–2.5) × 10$\mathrm{<mrow\; data-mjx-texclass="ORD">47</mrow>}$ erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. 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 suggest that the AGN obscurer is within a vertical height of 20 pc, and a radius of 125 pc, of the nucleus.