As difficulty adjusts automatically, it is indeed possible to run a mostly functional network with the bare minimum hardware required for storage and networking.

Naturally, this network will have next to no protection against a 51% attack, as anyone with a faster computer (or computers) will easily be able to outdo your bare minimum hardware.

Running it would be as simple as setting up a new node with a miner, and making sure it doesn't connect to the existing network - an easy way for that would be to have a new genesis block. You can then mine on this network, and the difficulty will adjust itself so that you have a 10 minute block time.