Effectively, players start with four separate tabs in each league, including events such as races. Items in the stash are separated by league, meaning a character in the standard league can't access the stash from the hardcore league, also, characters not in an event cannot access the Stash of those who are (ie: your Standard characters cannot interact with the Stash of those in the Delve Event).
Each stash tab is 12x12, for a total of 144 grid spaces.