In a super quick search:
H610 mobo $80
i3 12100 $130
Team Group 2x8 DDR4 $50
This is Newegg, but gives you a good idea of a start. You (could) go with the F skew for a savings but I don't recommend doing so. As an option you could consider one of the AMD G series but make sure to do your homework considering compatible mobo (in regard to BIOS).
You should be able to pick up something like a 1660 GPU for ~$200, or opt to look around at other similar performers. That would leave a tight budget for a case and an SSD to run it all on. At that budget you will probably have to consider a later update to more storage such as a HDD to store games on. Rosewill and DIY PC offer some workable cases for M-ATX in the $25-30 range. They aren't optimal but in (some cases) are better than a shoe box.
In a situation like this, don't overlook things like CL, Marketplace, LetGo and other similar situations. Buyer BEWARE with any of those but I recently sold a 2700X system with 1080 for $500 on CL.