LaCie is on the pricey end for RAID enclosures. They started off as a Mac-only company, and their pricing reflects that. On the flip side, that does meant their stuff is easy to use, but RAID enclosures are in general simple to use.
Sans Digital and Mediasonic make more reasonably-priced RAID enclosures. The 4-bay models are cheapest. 5-bays go up substantially in price, and 8-bay slightly more.
http://www.newegg.com/RAID-Enclosure-Subsystems/SubCategory/ID-509
For your storage needs and based on the fact that you already have 5x4TB drives, it sounds like you need at least 5 bays. RAID 5 protects you from a drive failure, but you lose the space of one drive. So a 4x6TB RAID 5 setup would only give you 3x6TB = 18 TB of storage. You probably don't need eSATA, as USB 3 is good enough for videos.
I agree with the recommendation to get two. RAID is not a backup, it is for reducing downtime - your files are still available even if a drive fails. You still need to backup the data to protect yourself against accidental deletions and overwrites, or file corruption. So two 5-bay enclosures would work. 5x6 TB drives in one configured as RAID 5 will give you 24 TB of storage. 5x4 TB drives in the other configured as JBOD will give you 20 TB of backup space. You may want to consider an 8-bay enclosure for the backups, that way you can increase your backup space by adding more drives (and it also gives you space to do things like backup your desktop and laptop, and there's no chance of you mixing up the RAID enclosure with the backup enclosure).
That's the easiest option. It's straightforward, and initial configuration is just inserting the drives and turning a dial or pushing a few buttons. The enclosure takes care of all the RAID and JBOD setup. Afterwards, you just turn on the enclosure and plug it into your computer via USB, and it acts as if it were a single external HDD.
A more elegant, and perhaps more useful solution would be to get a NAS. That's like an enclosure except with a small computer and network card attached. It costs a few hundred dollars more, but you can just leave it on all the time, plugged into your network. Any computer on the network can then access its files (provided you give it permission to access the files). Being it's also a computer, it can handle other tasks too, like handling your file downloads, organizing your photos into albums, etc. The downside is they cost several hundred dollars more than a plain enclosure.
The top of the line solution is to build your own NAS using a desktop computer. Use an OS like FreeNAS or Windows Home Server (deprecated, but had a lot of neat features). The upside is that since it's a real computer, performance will be much better. The downside is that since it's a real computer, cost and power requirements will be higher (if you're in the U.S., each Watt translates into about $1/yr of extra electricity for a device left on 24/7).
We don't know your budget or future requirements, so tell us which of these solutions most interest you, and we can provide more details.
Edit: Assuming all 5 of your 4TB WD drives are sitting in your desktop PC, you could just get a single 4-bay RAID enclosure and 4x6TB HDDs. Put the drives in the RAID enclosure, configured as JBOD. That'll make it look like a big 24 TB external HDD. Use this to backup the drives in your desktop PC. This would be the cheapest (if least-flexible) solution.