Step 1: Installing Steam

As the name suggests, the first step is to officially install Steam from Valve's site.
After letting it download and install everything it needs to though, you'll only see an error about OS compatibility:

This can be fixed by telling Steam to download an older client, the last Windows 7 one.

You may be tempted to just enable VxKex, but DO NOT DO IT! It will break running some games, some can be fixed with VxKex, some can't. The last Windows 7 client uses the latest Steam API anyways, so there's no reason to do so.

Step 2: Downloading the Windows 7 client

Open a command prompt window in the Steam folder, and run the following command:
steam -forcesteamupdate -forcepackagedownload -overridepackageurl http://web.archive.org/web/20240918104445if_/media.steampowered.com/client -exitsteam -textmode
(you may need to run steam -clearbeta beforehand)
You will see a client update window, just like before. This time, it will download the Windows 7 client.
After that, you can open Steam and see it works.

Step 3: Disabling the updater

Although Steam won't update itself to a broken version, I always like to do this on any client other than the latest one. Besides, it's not going to do much good here on Windows 7.
To do this, go to the Steam installation folder and create a file called Steam.cfg with the following contents:
BootStrapperInhibitAll=enable
BootStrapperForceSelfUpdate=disable

Step 4 (optional but recommended): Disabling GPU acceleration

If your Steam client takes a long time to actually become responsive, you will need to disable GPU acceleration.
Go to Steam > Settings, Interface and disable "Enable GPU accelerated rendering in web views (requires restart)"

And that's it! Enjoy!

Thanks to lightwo's blog for some information about this.