I am looking for a place to store my luggage in Taipei at a safe place, for about 2 weeks. I think it can be pretty hard to find something as I hardly speak any Chinese.
From the link provided in the comment by @pnuts, good places to store baggage in Taipei include,
Taoyuan Airport provides 24 hours luggage service. No matter whenever you arrive Taoyuan Airport, you can use the service. There price for the following list is for each time, up to 30 days.
Most major Railway Stations provide baggage service, including Taipei, Kaoshung , Taichung, Shifeng, Ruifang, Hualine. The service hour is 7：00 -22：00.
More information regarding pricing can be found by visiting the link.
In case anyone is interested:
I have now been there myself and have found out that it is possible to store your luggage everywhere where these companies are that also send luggage arround.
You will find it on this map http://www.taoyuan-airport.com/uploads/ap_documents/1030819/map_en.pdf, page 9, no 3 & 4.
I recommend that you use lalalocker, the link is https://www.lalalocker.com/en/ You can book online and store your luggage in nearby cafes, hotels, bookstores, clothing stores, etc.