In Taiwan, even when TRA trains are full, you can buy a "no seat" ticket. It costs 20% less than the regular price.
It means that you have no guaranteed seat, but if a place is free you can sit until someone arrives with this seat reservered.
Trains are quite spacious and each car comes with an empty area (it is labelled "keep clear" but sometimes it gets filled with luggage) where you can hangout and seat on the floor.
Curiously, a train "fully booked" (with only "no seat" tickets available) is not necessarily packed. Taiwan railways network is circular, it goes around all the island with plenty of stops, and a lot of passengers use it for small journeys. Between Taipei and Tainan for instance, there are around 60 stops (even though most trains won't stop everywhere). So if someone buys a ticket from the 24th to the 25th stop, the seat won't be available to go from Taipei to Taiwan, but possibly empty during 97% of the trip.
But keep in mind that trains can get packed (especially during the weekends and the holidays), so be prepared to spend the whole trip standing up if you show up with "no seat" ticket.