It’s perfectly possible to install and run Tvheadend as a single-seat installation, with the software running on the same system as any client software (e.g. Kodi), with all files stored locally.
Alternatively, you can run Tvheadend on a server, perhaps on an always-on system that houses your media, perhaps on a dedicated low-power system - it’s your choice.
Where you have aerial/coax connections might influence your choice - unless you use SAT>IP or have some other way to transport your TV signal over a LAN, your Tvheadend installation has to live where you can actually connect your tuners.
Wherever you install it, Tvheadend primarily runs on Linux - pre-built binaries are available for most Debian-based distributions (Debian itself, Ubuntu, Mint…) and RPMs for Fedora, or you can build it yourself. It runs on both 32- and 64-bit x86 and ARM processors, and so also can be built for Android (which uses the Linux kernel).
You will only need c. 30MB disk space for the application and associated files, and maybe anything up to 1GB for your configuration - depending on how many tuners of what type you have, how many channels you receive, your choice of programme guide, and so on. You’ll clearly need much more for your recordings, though: as a guide, an hour of SD MPEG-2 video will take about 1GB, while high bitrate HD H.264 will easily consume 5GB+ per hour.
Tvheadend is intended to be lightweight, so it will run on a NAS or similar low-powered CPU. Note that the exception here is transcoding: if you want to convert high-definition video in real time then you will need a powerful, multi-core system. It will happily run in less than 1GB of RAM, and many people run it successfully on original Raspberry Pis with perhaps only 256MB of usable free memory. This does depend on what else you’re using the computer for, though, as a GUI will drain your system as will any serious file serving.
And, of course, you’ll need one or more TV tuners if you want to receive regular broadcast television - otherwise, you’re limited to IP sources.
An Internet connection is recommended but not essential. You need to
have an accurate clock for EPG timers to work, for example, but this can be
synchronised from the broadcast signal if you’re not in a position to use
ntp or similar.
How about light enough to run on a travel router? Take a look at this example from one of our users…