Welcome to the Apple community.
If you use the Apple TV for something else, turn it off or otherwise allow its turn itself off, any downloaded you have saved to it will be lost.
With such slow Internet speeds, you may wish to consider downloading content to your computer, and streaming it from there. You will have to wait until it is fully downloaded on your computer, but it will not be erased when your computer is turned off or you use it for something else.
Another option would be to consider upgrading your Internet service to something which is considerably faster than what you have got now, perhaps 10 MB per second or faster.