Can a pet door only be installed on a door?
What are the other places to establish that might be suitable for a beagle?
In theory you could install a pet door anywhere, it's more a matter of what you're willing to do to your home and where you'd like the door to lead to.
You could install the door in an exterior wall leading to your yard, although that would be significantly more work and expense than installing it in a regular door.
Another option is to install a pet door in an outdoor kennel so that your dog is able to have an additional area to hang out in.
You could also install a pet door in an interior wall or door so that your dog could get from room to room without you having to leave full sized doors open. Again, this would require a bit of expense/work to do and it would look sort of silly.
Yet another option is to design a maze in your yard which your dog can only find his way through via pet doors. This would be costly and time consuming, but may be loads of fun.