I have several 4-5 TB drives that I usually just plug into my shield TV and watch shows and movies on. Is there a way I can just toss these into a big enclosure so it shows as one big drive and I can plug into my shield?
Generally, a NAS is the simplest way to accomplish this goal. However, if you want to use existing drives, they will have to be formatted in the NAS, therefore you will need to copy the data off at least one before you can start the process.
A 2-bay NAS is a great starting point and relatively inexpensive. It also allows for other future uses, backups, etc that may not be important now, but may be in the future.
Depending on the number of drives you have though you do have to decide what the best scenario for you is. A 2-bay NAS with a NEW higher capacity drive may be the same price, or cheaper, than a *big* multi-bay unit.
Keeping in mind that a NAS is a standalone device on the same network as your shield, this may not necessarily be what you are asking about. Assuming also that all these drives you care talking about are usb external drives already, if you are currently plugging them in to you shield. If that *is* the case, then a usb hub may be cheap and easy. However, someone else would have to answer if a hub is supported by the shield as I don't know, but suspect it is not.
As your needs are pretty minimal, a 3rd option, since the usb hub is likely not a valid choice, is something like a Raspberry Pi. Although it may be more work to setup and configured than a NAS, it would allow you keep the drives in the usb enclosures (up to 4 usb ports), assuming they are already, and create shares that the shield would access. This would be fairly inexpensive (sub $100), but wouldn't be my top choice to recommend.
I guess it comes down to how much effort and money you want to throw at the problem. A NAS would have the flexibility and power for sure, but is designed for internal drives. The Pi would require more setup but would allow you to use usb enclosures drives, but would not be as powerful.