There are some different ways to address this. Here are the main ones:
Raspberry Pi (my personal favorite)
As you said, you can attach your drives to a Raspberry Pi, and set it up to serve their content to the network. This is my favorite choice, since it isn't limited to what you want, but it allows a whole new world of possibilities (and learning!).
You begin attaching your disks to the network, and end up with a homebrew cloud-seedbox-backup server, and much more if you want.
There are a lot of tutorials on how to set up a NAS with a Raspberry, so don't worry if you don't know the details on how to do it.
I have been using one of these for the last year and a half, and my experience has been very good.
Pros: high customization, a lot of additional uses, not very expensive, really fun.
Cons: it's not plug-and-play (may take some time for the setup), you'll probably need a SATA to USB connector, transfer limited to USB 2.0 (max. 480 Mb/s) and Fast Ethernet (100Mbps).
Price: $35 (+ SATA-USB connectors)
Recommended if: you like learning and don't mind spending some time preparing it all.
NAS case (probably the best fit for you)
This is the commercial version of what you need. Right now there are hundreds of NAS solutions out there, that are specifically designed for your needs, without any hassle nor need of technical knowledge.
Pros: easiest choice, mostly plug-and-play.
Cons: most expensive solution, sometimes targeted for malware (see below).
Price: $60 - $800 (or more)
Recommended if: you want an easy, ready to use product, and you're willing to pay some more.
This is something quite arguable, but if I had one of these I'd be somehow worried about ransomware attacks that some users have suffered with their NASes. However, if you're not intending to connect it to the outer world, this shouldn't be a problem.
Wake on LAN
This isn't a full solution at all, but I thought it was worth mentioning. If your big PC's motherboard supports it, you could configure it so it would turn on when a magic packet is sent.
This way, you could turn on your computer remotely, using another computer, or your mobile phone (there are apps for that, just look for "Wake on LAN in the app store").
Using this method will only turn on your computer, so you would still need a way to turn it off when you're done transferring files, but I thought it was relevant in answer to this:
[...] which is a waste of electricity (and having to turn it on is annoying) [...]
Pros: it's not very difficult, and won't need additional setup (to what you already have).
Cons: doesn't fully address the problem, although you could turn off the PC with some script/automatic shutdown after X inactive time.
Price: Free!
Recommended if: you like your setup the way it already is, but your main concern is electricity, and automatically turning off the PC is a choice for you.
