You want a cheap NAS then go the AMD route. They have more cheap boards and cpu's that can deal with ECC memory. Also those Avoton boards from Asrock are nice but yeah a bit on the expensive side.
Also the 1 GB per TB of data is a myth and depends mostly on what kind of storage you have and how often you use it. I currently run a 80 TB ZFS server on 16 GB. It takes about 4 days on a LAN to fill the memory up and it serves the data through a Gbit port continuously, but I mostly have big files.
That said I would say 8 GB as a minimum though and just get them in 8 GB sticks so you can expand easier.
Also I personally run ZFSguru here but FreeNAS works well too so run what you feel comformtable with. As for a case. Just get one that fits your needs I had to make mine from cratch in the end as I move it to some LAN Parties as well and well 4U racks tend to be freaking heavy ;)
As for drives... I run a set of Greens (can be used without a hitch IF you run WDidle3 and disable or expand the headparking time.) the original 7 seconds will destroy your drives fast. A second set are a bunch of Hitachi 5K4000 Those are awesome but nowadays impossible to find :( My last set are a Bunch of WD Reds and they work great as well. I personally don't like Seagate drives as I had numerous issues with them in the past.
Lastly about hardware... IBM M1015 SAS card is about the best expansion card you can get. Cheap and good and if you want to big the HP SAS Expanders work great with them.
In terms of Raid/Pools if you go the ZFS route do note the following for optimal pools usage however it's not mandatory.
RAID-Z: 3, 5, 9, 17, drives
RAID-Z2: 4, 6, 10, 18, drives
RAID-Z3: 5, 7, 11, 19, drives
enable 4K sectors and use v28 or V5000 when making the pool.
Another note you CANNOT expand a made raidz with more drives. This is the main drawback of ZFS. You need to decide beforehand how many drives you should use. So you cannot buy 3 drives and later decide to add 3 more to the same raidz. you need to make a second raidz for it to work. You can however expand the drives. so you can upgrade from 2 to for instance 6 TB drives later on. Just pull one out plant the 6 TB in let it rebuild and do it with the next and repeat. Slow but works like it should.