I’ve encountered a few instances where my game would crash when trying to save the game state and a browser refresh would be required to access the game player again. The work around is to simply pause your game before saving. Not sure why this happens, seems to be specific to N64 games as well.
Again, if your game crashes when saving the state for N64 games - or any other game for that matter - pause the game before clicking the save state button and it should download without issue.