Two screws hold the stock to the receiver and barrel of most modern production bolt rifles. A sinlge nut secures the receiver and stock to the barrel of modern production pump and semi auto shotguns. The nut on the shotguns can be removed by hand on most models, but the screws for the rifles will require a screwdriver of some sort. In either of the above cases, disassembling the firearms into two shorter pieces takes about a minute. Disassembly reduces the length of these long arms by about 40%. As mentioned above, break action rifles and shotguns are typically take down guns, but that is not a hard rule. I had a NEF .410 single shot break action shotgun that was not a take down.
You will be hard pressed to get a break action gun (over-under or side by side) for $400, but Remington has a line of shotguns made in Russia called Spartan that are right in that range or a bit more. You may be able to get a Charles Daily for about that or a bit more. How far is it to Hamburg from Harrisburg? Cabelas generally has a good selection of used arms and you may be able to find one in that range. Local gun shops also usually have used guns for sale.
It would help if we knew a bit more about why you need a take down, or why you need a shotgun, or why you need a rifle. Any info about use would help in making recomendations.
I just thought about it, but Thompson Center makes the contender and encore rifles, and you can get a piston grip for the receiver and shortened barrels so that you can have a pistol that shoots rifle cartridges. Single shot break action, but smaller than any take down.