We demonstrate the existence of metamaterial blueprints describing, and fundamental limitations concerning, perfectly reflectionless omnidirectional electromagnetic absorbers (PR-OEMA). Previous attempts to define PR-OEMA blueprints have led to active (gain), rather than passive, media. We explain this fact and unveil new, distinct limitations of true PR-OEMA devices, including the appearance of an "electromagnetic horizon" on physical solutions. As practical alternatives we introduce alternative OEMA blueprints corresponding to media that, while not reflectionless, are nonetheless effective in absorbing incident waves in a manner robust to incident wave diversity.