I purchased an enclosed rack like this many years ago, but I'm glad I never used it. They are a pain to cool properly. Most of these types of rack will not come fully enclosed, with the front or sides being steel but with air holes for good airflow. The one I purchased was a StarTech.

Racks are usually counted in height by U (unit) which is a standard server height, so look for something in the 12-25U range. https://en.m.wikipedia.org/wiki/Rack_unit

I believe the one I purchased was a 32U with wheels and it was around 1500 USD plus freight (it arrived on a pallet).

What you're likely looking for is a wall mount rack. Wall mounted racks are going to be the same width (usually 19" 2 or 4-post) but will vary by depth. You won't be able to put a full length server in a wall mount as it's designed for half-depth network equipment, but should serve your stated need quite well. 19" shelf mounts are available too.

Some examples: https://www.startech.com/en-us/server-management/rk1232walhm https://www.startech.com/en-us/server-management/cabshelfv

Server rack? Wall or floor mounted, there are also rolling cabinets.

There are some who repurpose some Ikea-stuff as server racks: https://duckduckgo.com/?q=ikea+server+rack&iar=images&iax=images&ia=images&iai=https%3A%2F%2Fi.pinimg.com%2Foriginals%2F81%2F84%2Fab%2F8184ab11849051f4fa0f66f50b0ac97d.jpg