Try to find an used Intel NUC with i3 or i5 on ebay. You should get older ones (for example NUC DC3217IYE or DC53427HYE) with a bit of luck for about 100 to 140€.
Cheaper alternatives (for about 50 to 90€) would be an Intel NUC or Zotac Zbox with Celeron or Pentium.
But then you could have slowdowns here and there at some games.
Try to avoid systems with Celeron N or Atom Processors. They are to slow in my opinion.