Ansible-NAS will run against any x64 Ubuntu box (i.e. not a Raspberry Pi or other ARM hardware), or even a VM if you're just testing things out. The hardware you pick for Ansible-NAS depends largely on what you intend to do with your server - and is very much a "how long is a piece of string" type question.
Questions To Think About
What will be taxing the CPU(s) on your Ansible-NAS box? Are you intending to transcode video? How many users will be hitting it?
How many applications do you intend to run in parallel? How much memory do these applications require?
Are you going to run the ZFS file system? (you should!)
What are you intending to store on your Ansible-NAS? Is it data you can download again, or is it important to you that you don't lose it? Do you need mirrored disks?
Ansible-NAS development is tested against an HP Microserver N54L, with 16GB of memory, a 60GB SSD for the OS and 4x2TB WD Red NAS drives for storage. It works great :-)
This is obviously not the only solution but a reasonable one if you just want a single box to buy, and many different models are available on eBay for varying costs.