Depending on how tight your budget is, an Intel NUC might fit the bill. I have no clue, though, if their GPUs support 4k. But they are small, silent, and cover

Depending on how tight your budget is, an Intel NUC might fit the bill. I have no clue, though, if their GPUs support 4k. But they are small, silent, and cover a relatively wide range of performance. Low-end models start at € ~120 (NOT including HDD/SSD!)