How hard is it to become a video game story writer?

I've been writing for games for a couple of decades. I've written for PC, console, MMO, Facebook, and mobile games, so I have a range of experience. The short answer is no, you don't need a degree in programming. You need to be a very good writer with a good technical mind.

Yes, it will be hard to break in, but that's true of any creative field. It takes a blend of training (being highly trained as a writer), persistence, determination, and a bit of luck.

What you want to research is Narrative Design. This has become the dominant job for someone who is creating and implementing story in games. My definition of a Narrative Designer is someone who meshes storytelling with gameplay. That means you need a good grasp of storytelling and writing, as well as a good grasp of the basic principles of game design.

To get a sense of what training you'll need, I recommend that you go to gamasutra.com and search for Narrative Design jobs. Look at the qualifications that companies ask for. These days, they usually want a B.A. or equivalent experience.

It can't hurt to learn some programming, if you're so inclined, but it's never been an issue for me. I don't do code. Generally, companies are more interested in your writing ability, so having excellent samples of interactive writing is your goal. That's what you need to strive for.

There are a number of good books out there about Narrative Design, including mine: Writing for Animation, Comics, and Games. You can read a few of those books to learn more about the field.

Good luck!