If that's what you need, I'd consider looking at used Lenovo ThinkCentre Tiny PCs on eBay.
They are very small so they will fit in your media center no problems, and they should have more than enough power for your limited application. Although a used model will be a few years old, they should still be powerful enough for your needs. You can't build or buy a new compact PC for anywhere near the price of these used ThinkCentre Tinys.
You could also get an Asrock Deskmini, but they ship bare bones so you'd have to buy memory, storage and processor for them. You can buy a used Think Centre Tiny (granted an older/lower end model) for less than the price of a CPU to put in the Deskmini.
Intel NUC is an option others have suggested, but frankly they are a bit pricey compared to the alternatives.
If you want to entertain the Deskmini route I can make some more suggestions regarding the components.