First, what Mac OS are you using?
Secondly, what exact Mac Mini model do you have? I have a late 2012 model, and I do not have such issues.
Third, when you say "storage", but the title of your post is "memory", what do you mean?
Fourth, emptying the trash just cleans up some disk space, You actually need to perform disk cleanup just about every day, but getting rid of stuff you no longer need (like deleted EMails, prior versions of software, etc.). And, you need to do disk maintenance/repairs on a regular, periodic basis. I do it once a week.
Next, for any third party software that you are using, are you keeping such programs up to date?
Finally, are you making any backups to an external device? If so, what software do you use for that, and how often do you make backups?