I made a big mistake, and deleted my entire cache folder from my abantecart via ftp.
Have I completely ruined my installation now? I can't access anything via web, including the admin panel.
Is there any way to rebuild the cache?
You do not need to rebuild cache. It is rebuilt automatically.
Make sure you have ../system/cache directory present and this directory have writing permissions for your web user.
You can sett it to 777