We refer to this again as Champagne Cellaring or simply cellaring, probably because most people do it in their cellars. Before modern refrigeration and cooling, it was always done in cellars.
Even Non Vintage Champagnes benefit from a year or two or even more before drinking, and do many other wines.
Cellaring, long term storage so that wines age gracefully and positively, is more demanding than mere storage. You might store bottle so you always have some available, but you cellar to develop and age the wines gracefully.
Bottles must be protected from light and heat. Occasional light, such as when looking at the bottles is fine, but constant strong sunlight or fluorescents etc. is not. Humidity is also important in the long term so that the corks do not dry out. This is aided as well by storing the bottles lying down so that the wine is in contact with the cork.
Before cellaring, I prefer to purchase my bottles on or close to release from a reputable retailer to minimize the possibility of even slight heat damage, which will be amplified as the bottles age. I also do not like buying during a hot summer or after a hot summer, as often not much care is given to the bottles during transport.
That said, I do buy older bottles from some retailers (like The Rare Wine Company). Also, many people like my friend Paul has had lots of luck buying great bottles at auction, and I may eventually explore that route as well.
You really have three practical choices if you want to cellar Champagne, and you had best realize that Champagne is the most delicate of all wines.
You can storage your Champagne in a self contained freestanding wine cellars. I have a small one that resembles a small dorm room refrigerator, and you can also buy large ones which can be fairly elegant pieces of furniture. Mine holds perhaps a couple dozen bottles of wine, but ones that hold hundreds are also available. Do note that Champagne bottles are bigger than most wine bottles so your wine cellar will hold fewer bottles than advertised. Click here to explore freestanding wine cellars at Amazon.
You can also use the services of a commercial (temperature controlled) wine storage facility. In some urban areas you might have multiple choices, in some like Boston once choice, and in some areas non at all!
Of course you can also build a wine cellar in your home or condo (typically in a cellar/basement, but it could be in a closet, spare room, or other space). I recommend How and Why to Build a Wine Cellar, Fourth Edition by Richard M. Gold, which is both informative and an amusing read. I used it and had a contractor build a cellar based on my input from the book in my previous residence. There are plenty of contractors who specialize in wine cellars as well.
You may have a passive cellar, meaning with no cooling required. I was hoping that mine could be passive as it was built in the cellar on bedrock and had lots of insulation, but it warmed up to about 66 degrees in the summer. That's not a catastrophe, but less than 62 degrees Fahrenheit (16.7 degrees Celsius) is better (some would say under 60 degrees Fahrenheit, 15.5 Celsius). Slow seasonal temperature variations are not a problem just as the cellars in Champagne have, but you do not want frequent nor rapid temperature swings. Some people claim that mild seasonal temperature variations are actual good for wine!
If your cellar gets too warm or has other temperature issues like frequent or rapid variations you will need a wine cooling unit. Most people set these at 55 degrees Fahrenheit (about 13 degrees Celsius). Detailed descriptions are beyond this article, but wine cooling units are rated by cubit feet of cooling capacity. There are both self contained units (similar to residential air conditioners) as well as "split systems." You cannot use a standard air conditioner as they do not have accurate temperature control and more importantly will not maintain reasonable humidity, needed so that wine corks do not dry out.
Of course you will also need wine racks which can be custom built, or more economically bought from several commercial choices. I chose some wooden redwood racks that needed assembly. Be aware (once again) that Champagne bottles are larger than standard wine bottles. I stored mine both on top of my racks and in special Magnum bottle racks.
Wine and Champagne cellaring is worth it for many of us. Having a great selection of aged Champagne and other wines in your home is a joy!