Look up DLNA. It’s a system for streaming digital media between devices. It is likely that your smart TV will have a DLNA client.
Your router may provide a DLNA server for an HDD plugged into a USB port (if it has one; many do, e.g. Belkin, Technicolor).
Or you can buy any of the MyCloud family, and use the Twonky DLNA media server that is built in to it.
If your smart TV is based on a fairly open version of Android (that allows you to install apps from the Play store), you may be able to install Kodi and access a file server, again, either provided by your router, or by a MyCloud.
Yes, it is possible via WiFi, provided you have a router that provides wireless access.