Crabbie's is not really beer, it's really a soda pop with alcohol. You would have to treat this like homemade soda pop. You can purchase one of those home root beer kits that comes with the special CO2 relief caps to allow natural carbonation -- mix up a ginger ale soda pop recipe, add a little vodka and yeast, then refrigerate as soon as carbonated to halt fermentation. Or you could do the same but skip the yeast and special caps, and just force carbonate in a keg.
Bottom line is, you can make this at home, but it is NOT beer, it is soda with vodka. Seriously. I'm not kidding. No joke. Good luck. I might even want to give this a try myself as I do own a root beer kit as described above (which makes AWESOME root beer, by the way, hard or soft).