Reduce render file size in After Effects I am making a simple travel map, like the one in the Indiana Jones movies. I followed this tutorial:

                
               (German)

When I want to save it as a video (render it), the video file is 5GB huge. However the length of the movie is only about 1min. I tried to reduce the resolution to a quarter, then the file is still 700MB large, but with very bad quality.

Should I use a different program to do this task?
You need to change the encoding settings. By default, AE uses the AVI animation codec, which is very good at retaining quality but creates incredibly large files. Using an intermediary codec like Apple Prores, Avid DNxHR, or GrassValley HQx would give you visually equivalent quality with a significantly smaller file size.

If that file size isn't small enough for you, then you may want to go to a compressed codec like h.264. Choosing h.264 with a bitrate of around 20-30Mbps would give you pretty good quality. The disadvantage of going to h.264 is that you'll lose a lot of visual information, making any additional editing such as color grading more difficult. You'll also be using a multi-frame encoding which makes scrubbing through an editing timeline very difficult.