Generally, you repair tubes, not tires. From your last sentence, it sounds like you are indeed talking about the tire.
As it is with the tube, the real answer is "Size Matters". In this case, both the size of the hole and the size (okay, type) of the tire. If I was looking at damage to a road tire "slick", the hole you describe would likely having me change tires. On a cross or MTB tire I might try a repair. In addition to using a tube patch, I'd also glue the "L shaped hole" back together with superglue.
If there's any safety doubt, change tires. The closer to the side of the tire, the higher chance this could cause a lack of integrity while riding.
Never hurts to get a second opinion at your LBS either. Smaller shops will tend to give you a straight answer rather than just going for the sale of a new tire.