From a hardware perspective, you should have no problems with reasonably up to date hardware.
You might imagine that 8 tuners would require massive processing power, but in reality it is not the case.
If you are building a box from scratch, you could possibly aim to build a fairly low noise box.
A current Intel Core I3 would be up to the task but a Core i5 would be better. Get the best processor that your budget allows though. Your second priority should be compatible memory (some particular memory clock speeds will match better with the chosen CPU) and the appropriate mobo with a compatible sockets for your CPU and memory. A mobo with a HDMI output might be a requirement if you are using the box to directly display media to your TV. You should not be adding a graphics card because it will not be needed, unless you are using the box for gaming. It will only make the system noisier. Likewise an AMD Ryzen APU is probably not worth considering if you are not gaming because your investment will not be directed to where it's needed. You could consider a board that accepts m.2 SSDs for the OS as it will seed up boot time and improve system speed overall. A 240GB m.2 drive is more than enough for Linux and won't break the bank. Then a standard 3TB sata (or more) drive can be used for your recordings.
So in summary:
An Intel i3 to i5
Compatible memory (8 to 16GB)
A motherboard with the correct socket for your CPU, onboard graphics / HDMI out, m.2 socket and enough compatible PCI slots for your tuner cards.
A 240GB m.2 drive
A sata drive >= 3TB
A PSU (Bronze rating or greater). Modular types, mean only the cables you need are plugged in.
A compact case, one with good noise dampening could be a consideration.
With respect to streaming, it is possible to use a TVHeadend client such as Kodi on external networks by port forwarding your TVH server through the firewall of your router. Google "port forwarding TVheadend" for more info.
Transcoding video to stream is a different kettle-of-fish, I personally wouldn't recommend this unless you are on a limited data plan and have no choice. In that case, you will probably need to increase the CPU specs to i5 and i7 processors as transcoding is very processor intensive. To be honest, with full load on your tuners, you might struggle.
I personally use Ubuntu, it's got a good support base.
Sounds like a great project. My own box is due an upgrade soon. I might compare notes. Good luck with it.