We present comprehensive multiwavelength observations of three gamma-ray bursts (GRBs) with durations of several thousand seconds. We demonstrate that these events are extragalactic transients; in particular, we resolve the long-standing conundrum of the distance of GRB 101225A (the "Christmas-day burst"), finding it to have a redshift z = 0.847 and showing that two apparently similar events (GRB 111209A and GRB 121027A) lie at z = 0.677 and z = 1.773, respectively. The systems show extremely unusual X-ray and optical light curves, very different from classical GRBs, with long-lasting, highly variable X-ray emission and optical light curves that exhibit little correlation with the behavior seen in the X-ray. Their host galaxies are faint, compact, and highly star-forming dwarf galaxies, typical of "blue compact galaxies." We propose that these bursts are the prototypes of a hitherto largely unrecognized population of ultra-long GRBs, which while observationally difficult to detect may be astrophysically relatively common. The long durations may naturally be explained by the engine-driven explosions of stars of much larger radii than normally considered for GRB progenitors, which are thought to have compact Wolf-Rayet progenitor stars. However, we cannot unambiguously identify supernova signatures within their light curves or spectra. We also consider the alternative possibility that they arise from the tidal disruption of stars by massive black holes and conclude that the associated timescales are only consistent with the disruption of compact stars (e.g., white dwarfs) by black holes of relatively low mass (