There are very cheap routers out there with usb ports, (eg. TPLink WR703N) your current router might even have one, you can plug an drive right there, they usually include some kind of software or you access it by the right ip address.
There are a ton of NAS solution out there from the WD mycloud dual bay to more slots, those being very plug and play. You can also get any old dumpster computer and install freeNAS on it, very easy aswell, alot of tutorials out there. You can ofcourse enable access over the web but if its trivial for you then just don't do it so that you don't have to worry more about security.