You can extract zip and rar files in Synology DiskStation DSM. However if you have multipart archived files like r00, r01, r02 etc. you will run into not being able to extract the files.
The reason is simply that it does not know the file type. To solve this and extract the files you need to find the .rar file. Try searching for the filename.rar ie: really-bigfile.r00 search for really-bigfile.rar. Right click and choose extract.
The problem is the the paging. File station, as default, only show 50 files on each page. Due to the default sorting, the rar-file will be shown on page two or three. To find the file, jump to the last page, or change the number of items shown per page.