From GameWatcher: “Apex Legends has no shortage of various ordinary and less-ordinary errors for players to contend with, though Code 408 harkens back from the days of Titanfall 2.
Apex Legends Code 408 is an error relating to matchmaking server connection or, rather, the lack thereof. It’s a bug that results in an immediate disconnect and unlike some others, a simple restart of the game may not be able to fix it every time. We do have a potential solution, however.”
Source: N4G PC Apex Legends Code 408 – What Does it Mean?