I'd like to know how many hours of programming one would need in order
to begin a viable TV network. For example, if I wanted to start my own
tv network, how many hours of content would I either need to produce
or acquire in order to make my network a viable, functional entity?

Watch a few TV stations or better yet, review a TV show listing and do
the calculation.

This depends on regulations that are different from one place to
another. It might help to know, where you'd like to establish the
network.

the network would be established in the USA.

Don't forget that whatever inventory you start out with, it constantly
needs to be refreshed.  Whether you produce your own content or
license programming from others, each show, film, series, whatever --
will have its own license period that expires at different times or
goes into a
blackout period.  So the inventory needs to be steadily "fed" to keep
a flow of programming going over weeeks and months.

And don't forget to factor in interstitial programming -- promos or
commercials running through your 24 hour programming day -- alerting
viewers what's on the network and what's coming up that night, that
week, etc.  This can fill up hundreds of hours of airtime a month.

You raise an interesting question.  UPN runs its network with about
300 hours of programming a year since it only broadcasts two hours a
day six days a week, and then repeats its programming.  I believe Fox
originally started with four days a week and expanded to five, six,
then seven.  UPN use to be only two hours five days a week.  The FCC
considers you a TV network if you have 13 or more hours of programming
a week.

If you're talking about a cable TV network, I believe the Home &
Garden Network started with 100 hours a month that they then would
repeat over and over as they built up their programs.  It helps if you
can tap a library of some sort for your programming.  For example, if
you decided to do the Silent Movie Network, you would then go out and
contract with everybody who has silent movies.  The Game Show Network
was a great idea because there are tens of thousands of hours of game
shows collectiing dust and not generating any income, so you could get
them cheap.  Good luck.