There are a lot of threads in discussion forums which ask how much money is the bare minimum to get by in Taiwan. Since many countries now get up to 3 months of visa-free entry, there are a lot of backpackers looking to set up shop but not set their roots too deep, and students who are here for a short exchange. If you're in the 1 to 9 month category, this guide will prove to be best suited to you.
The simple answer is if you plan to be in Taipei you will need 15,000 NTD or around 500 USD per month. If you live on the east coast or a smaller city you can get the number down to about 12 to 13,000 NT a month. Food and entertainment prices are generally the same, but your rent will be lower and quality of life a bit higher.
Here's how it works out:
Unless you're joining family, have a significant other, or some secret way to live for free, you'll need to pay rent somewhere. If you're staying in Taipei or New Taipei City for less than one year it will be hard to find an apartment. The bare minimum most places will negotiate is six months, and your rent will be insanely high. In smaller cities it's easier to find places without huge contracts because there is more availability.
There are a host of single unit apartments near the universities, but they're isolating and depressing even at their best. Smartest advice is to sink up to about 8,000 NT into a long term stay at a hostel. A quick search of the internet will even help you find places which specialize in long term accommodation. They have kitchens, a stable set of residents, and usually provide fairly safe places to keep your valuables.
As a budget conscious short term stay person, if you pay more than 8,000 NT you're wasting cash.
Rent: 8,000NT or less.
For foreign visitors the food in Taipei's night markets look cheap at first glance. There are noodle shops and restaurants which offer reasonable meals for 50 - 150 NT. If you eat two low-cost meals a day you'll be at about 300 NT, or 9,000NT a month. But you can easily cut that in half and eat better tasting and healthier food.
Get a place (hostel especially) with at least a hot water machine, a gas or electric cooking range, and a frying pan.
In preparing for this write up, I went through my receipts for a basic tally of how much I spent on eating at home for a whole month:
3 pounds of oatmeal : 120 NT
20 eggs: 100 NT
5 lbs chicken: 400 NT
6 gallons milk: 1200 NT
8 Wellcome supermarket steaks: 400NT
5 lbs noodles/pasta: 200 NT
3 heads of cabbage: 90 NT
Other greens: 200 NT
Fruit: 700 NT
4 bags of frozen dumplings: 200 NT
4 fresh fish: 400 NT
Total Bill: 4010 NT.
All the major stores have buy one get one free deals and other discounts, so some of the prices above reflect this. You're going to be staying for more than a few weeks, so stock up on things when they go on sale and you can keep your cost of living low. Oatmeal or eggs for breakfast, meat + a veggie + fruit for lunch, repeat + carbs for dinner. If you're a vegetarian you can get by even cheaper, but for reasons not explained here you'll need to eat a LOT of different vegetables and protein sources to keep your nutrition up.
Not on my list is rice. It's a pain to cook and clean it, but you can get six pound bags for about 150NT. I don't use pasta sauce, instead cooking veggies, an egg, and the noodles together in a soup like Taiwanese folks do.
If you live in a city where the service exists, you should be using the U bike. Assuming you ride everywhere in 30 minutes, which is for the most part reasonable, you'd be spending 10 NT a day on a round trip to wherever you were going. Assuming 2 round trips (or 4 rentals) per day, it's 20NT. So 600 NT per month for U bike rides. If you will stay for longer than 3 months, buy a used bike or a junky new one from a big box store. Including lights and a helmet it will set you back 2500 NT or so. There is an intro to biking safely in Taiwan here.
The bus and MRT systems will kill your budget, and should only be used for really long trips like Xindian to Danshui (tamsui), crossing Kaohsiung, or similar. The bike paths along the river in Taipei are really nice, and it's way cheaper to ride a bike than the subway. In smaller cities you're not only wasting money but time by using most bus services.
Avoid short bus and MRT rides like the plague. Assuming you're able bodied there is no reason you can't walk short distances. From Taipei Main station you can walk on average two MRT stops in about 15 minutes if you walk at full pace.
Total monthly transit costs: 1,000 NT or less if you end up buying a bike and staying longer.
If you want to slide by on 15,000 without using shortcuts, you'll only have 2,000NT left for eating out, drinking, or other activities. Remember if you don't have shampoo, a blanket, etc., you'll need to buy all of that when you arrive.
Beers at bars: 100 NT and up - average is 175.
Beers from stores (six pack): 110 NT to 300 NT.
Hard liquor: 250 NT and up.
Entrance to clubs: Girls: Free to 800 NT. Boys: 500 to 800 NT is normal.
Museums: 10 NT to 300 NT.
Movies: 190 NT to 320 NT.
Clothes: 98NT and up.
Coffee at a sit down coffee shop: 70 NT - 180 NT.
Pain reliever, basic medicines: 30 NT and up.
It's not glamorous to live on the bare minimum, so do yourself a favor and budget a bit more. Transit cost can easily double if you decide not to brave the typhoons on your bicycle and take the MRT instead. Luckily, Taiwan has a ton of free activities where you can socialize, practice Chinese, and be active without spending cash, so don't stay home alone just to save cash!
The lowest cost I've managed to live a fulfilling Taipei-based life on, including some meals out, a bit of nightlife, and a movie or two is 14,000NT, including rent.
If you think that's too low, remember the average starting salary in Taipei is 24,000 NT a month, and somehow local folks get by just fine, even living on their own. The whole list above also explains why Taiwanese people are famous for having all their money saved up for a rainy day, so, get saving!