It's not possible.  Such an enumeration could potentially have an uncomputable order-type, possibly equal to the order-type of the rationals.  (In other words, there could be countably infinitely many elements between any two elements)

It's possible to define a computational system where you can do arithmetic on countable ordinals, but it has the expressive power of Turing machines with oracles (where an oracle is a thing that correctly guesses the "right" answer for a computation that does not halt in finite time (consider a sequence approaching pi as a limit).   We can re-interpret the oracle's guess as passing to a limit ordinal.  In any case, TMs+ oracles are strictly stronger than just TMs.