To obtain Visitant Status, talk to a Conflux Surveyor. In the chat menu, you are able to select how many Traverser Stones you wish to trade for time allowed in Abyssea. If you have already obtained Visitant Status from a previous visit, and you have time remaining, you will also be able to use that time.
Initially, you may accumulate up to 120 minutes worth of time. Each Traverser Stone traded is worth 30 minutes. A total of 4 stones is required for the 120 minute total. If Visitant Status is allowed to completely expire, you will be ejected from Abyssea.