Problem with ATM
There are quite a few people that had issues with the ATM. I have no idea why, but here's what you can do to prevent this.
1) Go to your bank and ask
Before you leave for Taiwan, go to your bank on what to do if you want to withdraw cash in Taiwan.
2) Carry USD or EUR with you
How much USD or EUR you want with you is your choice. But there is an upper limit on how much is too much at the customs, anyone who has more than ten thousand USD upon entering should fill out the forms and abide by the regulations, or else confiscated and prosecuted, any currency equivalent to ten thousand USD as well.
On that note, valuable items such as raw diamonds would also be counted towards the ten thousand USD limit.
Airports accept changing USD or EUR. Withdraw from your account may be denied at the counter in the cities.
3) Only shop at somewhere that can use credit card
If you only have a credit card on you, then I'm afraid that your options are limited. Since most of the stores and shops will not accept credit card, it would be safe to solve this problem before you leave for Taiwan.
4) Online payment
Apple pay and other online payment apps are not widely accepted in Taiwan, just yet. It's even less likely to be accepted than credit cards. Some applications are accepted but not the others.
All in all, ask your bank before the trip or carry some USD or EUR might be your best chance of having enough cash on hand.