Xbox Classic PC build?

I've seen LinusTechTips build a PC into an Xbox classic case. I would like to do that, but only with hardware that I still have lying around here at home.

Now my problem is the following;
The motherboard from an old PC (I think it is Micro-ATX) just fits into the case, (I just put the motherboard on top of it) but now I have no room for a power supply.

I have already looked on the Internet, if there are cheap Mini-ITX motherboards that support a LGA 1151 or 1155, only they are priced at 60 euro + and I would like to spend for a craft project not necessarily only for the motherboard, especially I would like to run this as a retro console with all sorts of emulators.

The second option would be to install a smaller power supply, so that I can install the motherboard over the power supply and the free space dadrunter for hard disk, peripherals etc can be used.
The dimensions of the Xbox classic are:

Length: 40 cm

Width: 30 cm

Height: 10 cm

A normal power supply has a height of 8.5 cm. As will probably be (with RAM) no more space.

Does anyone have a solution to the problem?

L. G.

~ Chris

I would install a picoPSU and then use an external power supply (about one for notebooks).

For consoles, external power supplies are not uncommon.

picoPSU is such a power supply, because you give 12v in, and converts it all as you need it. Such a pico psu is very small.

Latte panda systems, are also some where you can install a real GPU, mini versions of such cards should not be the problem fits in any case in the housing