We have analyzed a number of intense X-ray flares observed in the Chandra Orion Ultradeep Project (COUP), a 13 day observation of the Orion Nebula Cluster (ONC), concentrating on the events with the highest statistics (in terms of photon flux and event duration). Analysis of the flare decay allows to determine the physical parameters of the flaring structure, particularly its size and (using the peak temperature and emission measure of the event) the peak density, pressure, and minimum confining magnetic field. A total of 32 events, representing the most powerful 1% of COUP flares, have sufficient statistics and are sufficiently well resolved to grant a detailed analysis. A broad range of decay times are present in the sample of flares, with τ_lc (the 1/e decay time) ranging from 10 to 400 ks. Peak flare temperatures are often very high, with half of the flares in the sample showing temperatures in excess of 100 MK. Significant sustained heating is present in the majority of the flares. The magnetic structures that are found, from the analysis of the flare's decay, to confine the plasma are in a number of cases very long, with semilengths up to 10¹² cm, implying the presence of magnetic fields of hundreds of G (necessary to confine the hot flaring plasma) extending to comparable distance from the stellar photosphere. These very large sizes for the flaring structures (length L R_*) are not found in more evolved stars, where, almost invariably, the same type of analysis results in structures with L ≤ R_*. As the majority of young stars in the ONC are surrounded by disks, we speculate that the large magnetic structures that confine the flaring plasma are actually the same type of structures that channel the plasma in the magnetospheric accretion paradigm, connecting the star's photosphere with the accretion disk.