Is there any in Taipei where the public has either free or paid access to?
Try the universities.
Alternately, there are loads of outdoor facilities and people to play with all over the city.
By the way, if you find an indoor place and want to buy a top-of-the-line indoor-only basketball (Spalding ZK Composite) used only a bit, private message me.
Life is hard, then you die.
There’s a semi-indoor one just behind my house, next to tennis courts, a playing field and running track. Its not really indoor, but its fully covered so you don’t get rained on, and it stays lit until around 11p.m. Has showers, etc. Anyone can use it – first come, first served. Trouble is, its way down in sunny Bitan.
It’s hard to find. Just ask your friends if they know of some friends. They usually rent the gym and have everyone pay before hand or after the game.