I'm assuming that you want to make the gun smaller for transport. Now, I don't have my 10/22 with me, and it has been years since I have actually handled one, but if I remember correctly, there are two screws holding the barrel and action to the stock. It would be easy enough to remove those screws and remove the barrel and action from the stock. As a practical matter, almost all modern firearms are "take down" arms. A single nut holds the barrel to the action of a pump or semi auto shotgun. Double guns are obviously take down guns, and two screws secure the barrel and action to the stock of modern bolt action rifles. The only firearm that I can think of that is not easily taken down is a lever action, but I'm sure that the buttstock is easily removeable. I've just never done it. I wouldn't pay for a "take down" 10/22, they (and almost all other firearms) are "take down" by nature.
Devilman, you cannot fire anything but .22 wmr from .22 wmr firearms. It is true that you can fire CB, short, long, and long rifle from a .22 LR. However, there is a difference in construction of the cartridges between these smaller rounds and the .22 mag. The diameter of the .22 mag bullet is fully encased by the case of the cartridge. Thus, the case is of larger diameter than the bullet. In the smaller offerings, the bullet and case are of the same diameter. The actual bullet has a smaller diameter stump that is encased by the casing. Firing one of the smaller .22s through a .22wmr pretty much guarantees a case failure. I hope this makes sense, but if it does not, then I recommend buying a box of each and comparing the different rounds. I do not recommend firing the smaller rounds through a .22wmr, however. It is true that the CB, short, long, and long rifle can all be fired from a .22 LR though. For those that do not know wmr stands for Winchester Magnum Rimfire.