Student or Learner
Do I need "a" here if the words "Christmas Eve" is not modified?
The story begins on (a) Christmas Eve.
It's more natural without the article or as "The story begins one Christmas Eve".
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
To my ears, it's fine - especially if it's "on a Christmas Eve in a small town in..." or "on a Christmas Eve that showed now sign of allowing a white Christmas" -- in other words, some other qualifier that says why this particular Christmas Eve is in some way distinctive.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.