Same reason why Linus put his gaming computer in a 4U rackmount chassis in the first place.
If he has other computers built into rackmount chassis and already has a server cabinet, he could consolidate all of them into one cabinet.
I looked on Newegg for a 15U server cabinet, and there's one from iStarUSA that'll run you USD $699. It has a perforated front door (not glass because then it will choke out the rackmounted computers).
There are various 4U tall rackmount chassis everywhere. I would recommend taking a look at this Rosewill 4U rackmount chassis that will fit E-ATX boards (12 x 13), has 7 PCIe expansion slots, and has a triple 5.25" ODD bay so that you can choose to install an iStarUSA 4 3.5" hot swap cage to greatly expand your HDD storage needs. Cooling shouldn't be an issue either. Plus it will take an ATX power supply.
For rackmount sliding rails, I would strongly recommend to go with the ones Linus uses for the server room and for his personal server setup. They are excellent, but I don't remember the brand he uses.
Well, the width and U sizes are all the same, it's how many Us high it is and how far back it goes that matter, servers need full sized cabinets that can cost hundreds for one. I'm not sure why you want a gaming rig in it though, but most 4U cases are either custom or from OEMs like HP and Dell. I know Cablematic make a cheap one, and Supermicro make ones but they mostly come with Xeon boards already installed. I think they do sell them on their own though. Glass front is not advised, server need to breath, so unless you have a lot of fans to put at the top and bottom don't.