Fileserver or NAS
If you ever worked in an office, I'm pretty sure you used a Windows computer and the company fileserver in the LAN and dreamed of setting up the same kind of perfect networking at home some day. Because it is so convenient - you have all your documents in My Documents folder and store all your videos, music, work files and whatever on your fileserver.
You can stream any file from your huge videos collection to the big TV set in your living room, you can stream any file from your huge music collection to your bedroom - or download them to your car in the garage, you can view your work files on your desktop.
Definitely you don't want a huge corporate server in a refrigirated room - but people write modern routers can be used as a full-fledged internet gateways! They can be fileservers as well, if you plug in a flash drive. Flash drives are slow and unreliable. You can buy a 1-2TB USB HDD and you will have a real server in your LAN! An internet gateway (with a front-end in DMZ zone) and a real fileserver!
Router + USB HDD
USB HDDs are affordable and can be connected to your router. You have USB 2.0 ports in your router - USB 2.0 can theoretically transfer data at a very high speed of 480 megabits per second (mbps), or 60 megabytes per second (MBps). The problem is, in practice USB 2.0 port of the router usually has a much lower speed, like 16 mbps or 2 MBps.
Router USB 3.0 port
If your router has a USB 3.0 port - that's impressive! Theoretically it can handle up to 5gbps (640MBps). What throughput it will have in practice is another question. You may try and find out. If it is half as good - that may be enough for your home needs. You will have a router hanging on your wall and a USB HDD hanging nearby.
How good is that?