Well.....let me just start with telling you there are tons of people on here with more knowledge than myself.
I make pulled beef all the time, but I generally only use the most fatty / marbled pieces of beef I can find for pulled beef.
Gerenerally a chuck roast of some sort around 5 pounds makes a nice cook for me...it takes most of the day and can be held for a few hours if needed.
I have used eye of round before as well.....
I used that cut for Italian beef sandwiches and other sadwiches that are similar....where you dont want to chew through fat on a sandwich.
A lean cut cooked to medium or less and then shaved thin makes for a nice eat on a sandwich.
I have never tried to pull a eye of round roast.....if it looked like this ( below pic ) then there may be issues with it when it comes time to pull....but again I have never tried to pull this. It may or may not......either way nothing will pull unitl you cook it to about 200 degrees.....
So if it does not pull it will be an overcooked piece of beef that I am not sure what you can do with what is left besides maybe soup or something.

Close up pictures of your meat........will help us advise.
