You have 4 options - Attach a USB HDD to your router and share it, Setup a SBC (like RPi) with a USB HDD, Buy a NAS and install one or more HDDs or setup a desktop for the same.
I would suggest the router only if you want to spend nothing. The limitations are with the router itself so YMMV.
On a budget, I would suggest the RPi with the USB HDD - this should do a good job as long as you are not going to 1. watch 4K rips with high bitrates AND 2. Multiple clients at the same time. Ensure you get one of the newer RPi's with more RAM, setup the same.
If you want to watch high bit rate 4K rips you should go for either NAS or (ideally) I would suggest a desktop which has multiple SATA ports so that its more future proof.
This should be a Pentium class or above (with integrated graphics) which has the transcoding capabilities needed if you need to stream to devices which do not support 4K.
Install Windows or Linux or whatever is to your liking and share the same for a simple setup. For a more better way, for ex. If you want to do transcoding and want an easy way to manage the same, setup Plex. The free account should be more than enough for your needs, as long as the TV is acting as a Cast target.
TL;DR - depends on what you are - simple and tight budget use - router with USB HDD, simple and money to spend - NAS, complex on a budget - Raspberry Pi, complex with money and you want the best of everything - PC as NAS. If you find the write up complicated, please feel free to ask.
