There are plenty of options to stay fit in Taipei. On campus at NTU members of IAA have access to preferential membership rates for the sports facilities. Private gyms are relatively common and many are open 24 hours. The great outdoors also beckons, with a dense network of cycle paths along the riverside, hiking trails in the hinterlands above the city, Yangmingshan national park a short bus ride away, and the rock climbing and sea diving areas of Longdong and Yeliu near Keelung accessible via bus and train.
As employees of IAA, we have access to the special NTU employee rates at the NTU sports center. Furthermore, there's an Olympic-size pool just south of the NTU sports center. It only costs 600 NTD for a six month season, so it's by far one of the most cost-effective means of exercise. The hours are:
Summer (May to Oct): Mon-Sat 06:00–18:00, Sun 06:00–11:30
Winter (November to April): Mon-Fri 06:00–13:00, Sat/Sun 06:30–21:00
Additional information (in Chinese) is here.
If cycling is more your thing, then renting a Youbike is a cheap and effective option both for transport around Taipei and getting some exercise. An extensive network of cycle paths exists along the rivers around Taipei and separate cycle lanes exist in many streets. To access the Youbike system you will require a Taiwanese mobile phone number and ARC for registration with the system. You can then rent and return bikes at any of the many bike stations dotted across Taipei (often closely associated with MRT stations) using your Easy Card (your NTU ID functions as such when you get it, or one can be purchased from most MRT stations). Unfortunately you can link one Easy Card per mobile phone number.
A popular past time amongst the locals and foreign national alike. Many trails offering spectacular scenic views are accessible directly from the terminal stations of the MRT lines in Taipei, or by short train rides. Trails are generally well maintained with paving, stairs (with bannisters) on steep sections, and ropes or chains to assist in traversing rocky outcrops. There are several very active social groups catering to hikers of all abilities in Taipei.