The old method of aging meat is known as dry aging. Dry aging is done by hanging meat in a controlled, closely watched, refrigerated environment. The temperature needs to stay between 36 degrees F and freezing. Too warm and the meat will spoil, too cold and it will freeze, stopping the aging process. You also need a humidity of about 85 to reduce water loss. To control bacteria you need a constant flow of air all around the meat, which means it needs to be hanging in a well ventilated space. The last and most important ingredient in this process is an experienced butcher to keep a close eye on the aging meat.
There are many reasons that butchers don't typically age meat these days. First of all the cost of aged beef can be very high. Because of the weight loss of aged beef, the price per pound can be pretty outrageous. If you add in the time, storage space, refrigeration, labor that price just keeps moving up. For aging to properly improve the quality of a cut of meat, it should contain substantial marbling. This means that there is fat evenly distributed throughout the meat. Only the highest grades have this kind of marbling and make aging worthwhile.
Because of the high price and the space necessary to age meat, dry aging has become very rare. Actually only a few of the finest restaurants buy aged beef. Many in fact, have taken to aging their own beef. This can be a risky job if you don't know what you are doing and I strongly suggest a good sense of smell to anyone who tries it. If your aged meat doesn't smell right, throw it out.
Aging takes about 11 days before you see much improvement in the flavor of the meat. After that the flavor continues to intensify, but so does the loss of weight and the risk of spoilage. Eventually the meat will be worthless so many fine restaurants who do their own aging will limit it to 20 to 30 days.