In February 2013 Steam for Linux was released. Back then, “Steam for Linux” meant “Steam for Ubuntu”. Installing and running it on Debian was not as hassle-free as it should be. But fortunately by now it is actually very easy.
dpkg --add-architecture i386
apt update
apt install steam-installer
And you are good to go
With Proton enabled in Steam it is even possible to play Windows games on Linux. My current personal favourite game - Satisfactory - is not even officially supported but (in Early Access) runs even more stable in an unsupported Linux environment than on Windows.
As I no longer have an Nvidia graphics card (they became too powerful and greedy IMO) the troubleshooting below might no longer apply as I cannot test it anymore.
If you are using XFS or any other 64bit filesystem on the partition where Steam and its games are installed you are out of luck. Somehow Steam does not like 64bit inodes, so running it on ext3/4 is mandatory I fear.
There is some reading material for more information on this. As far as I know the only game that is not affected is Kerbal Space Program.
I should have known this sooner, shrinking XFS to make room for an ext4 Steam partition is unfortunately not possible
If you get errors about Jockey missing, run this command (as root):
/usr/lib/steam/steam-debian-depends.sh
This will only happen on amd64 systems (which probably is the majority nowadays), an Nvidia card and you didn't use the official NVidia drivers. If you get errors about missing libGL.so.1 when running Steam, just install libgl1-nvidia-glx:i386:
apt-get install libgl1-nvidia-glx:i386
If it is not yet the case, you might have to add the i386 architecture:
dpkg --add-architecture i386; apt-get update
To query which foreign architectures are supported on your system run
dpkg --print-foreign-architectures