The fastest, cheapest, simplest thing you can do is put an internal drive in your PC and serve it up 24/7. Next up, the external USB drive; it is slower, less reliable, and cost more than an internal drive, but its still inexpensive, simple, and fast.
After that would be some sort of NAS. But I personally would take the extra money from a NAS and invest it in more drives for back up and redundancy and upgrade my network to make sure everything is gigabit..
And don't forget about how you will deal with failures. Internal and USB drives are readily available; NAS components are not as easy to source or could be more picky about hardware, while some network drive and router would definitely leave you limited on your options in the event of a failure.
As a note, a platter style drive will not typically saturate gigabit Ethernet at 125MB/s, and if your internet can, I wanna know who your provider is. Plus, like others have said, use link aggregation from your PC to your router (if you can get a router that supports it) to up the available bandwidth.
Also, an idling computer will only cost about $60/year per $0.10/KWH @ 70 watts idling (many are less these days) And since you'll have it on during the day anyways probably, that means you might be able to save a maximum of $20 a year or so (per $0.10/KWH)
A network drive and router will limit your options and scalability compared to your PC. Maybe you'll want to add a back up drive. Maybe you'll want to add another drive and setup a RAID 1 for redundancy. Who knows. Fact is a PC is pretty open on options, a network router and drive will be very limited on what they can do.