The original Xbox version of Conker's Bad Fur Day **does not have a manual save feature**. You cannot save mid-game. Progress is only recorded at the end of chapters or specific checkpoints determined by the game.
The Nintendo GameCube release (Conker: Live & Reloaded is Xbox; the GC port of the original N64 game is different) — actually, Conker's Bad Fur Day was N64 originally. The Xbox version is Conker: Live & Reloaded (remake). Neither the N64 original nor the Xbox remake has a traditional save-anywhere system. The N64 used battery-backed cartridge saves at chapter ends.
Accept the challenge; the game's humor and design are balanced around this limitation.