The reality is if you want something affordable you will have to look in the surrounding areas and not Taipei City itself. The MRT has expanded enough to service almost all of New Taipei City with many other areas soon to get access.
Yes, you will need a scooter or car or rely on public transport such as the MRT.
Everything on the MRT is accessible within a 1 hour ride, the exception being the airport of course. A 3 hour ride round trip is unlikely unless you are doing many transfers and its during peak hours. 2 hours max for an all day commute.
You can live cheaper with good quality housing and work at normal salary rates in New Taipei City. Then if you want the nightlife, shopping, etc. of Taipei, just head on in to the city. No need to commute into the city for work. Plenty of jobs outside the city. In fact, the housing trend now is for people to buy and live in Taoyuan, Sanxia, Tucheng, Linko and Shulin. Those are the hot spots and many new schools are opening up to service those people. Hell, Kangciao is hiring for their new location in Linko.
So yeah, it is crazy to live in Taipei City when there are reasonable alternatives.