How can I use Steam without connecting to Steam's servers?

I believe your problem can mostly be circumvented by using Steam's offline-mode.

You say you have the game on "your laptop". I assume this is a personal laptop you can take home from time to time.

Here's what you do. In your home, with unhindered access to Steam, download the games you want and start them once. This way Steam can verify your right to run those games at least once. After this, put Steam to off-line mode. It will remember the verifications you did (for limited time, a couple of weeks, according to @badp) and you should be able to use the laptop and its Steam installation to play the games without needing to connect to the Steam's server.

Depending on the game, you may even have the game online without needing to connect to Steam's servers.

Other than contacting the firewall's administrator and whitelisting Steam servers there's not much you can do. As suggested in comments below you can connect to Steam using proxies.

I know this doesn't solve your problem, but maybe it will be sufficient to someone else. I had the same problem with my University, where Steam logon server was blocked. A friend of mine found a workaround that surprisingly worked:

• When at home, connect to Steam and log in.
• Put your notebook in sleep mode.
• Power on the notebook at work.

Steam should be connected, games can be downloaded and you can chat with friends, but all games using steam servers (eg. Counter Strike etc.) won't be playable. Nevertheless, it's still more than you could do before (unless they fixed it somehow).

I had this problem in my University. I simply connected to my 3G on the phone and logged into Steam there. It doesn't consume a lot of data just to log in and authenticate your game to start. Don't expect to play online games though. Even if you have fast 3G/4G connection the latency will still be a problem.

Look on Google how to turn your phone into a WiFi hotspot according to your phone's model and system. Good luck with the games friend!

If you have access to a SSH session on a remote server, you can download ProxyCap or Proxifier, make a Socks5 using SSH via PuTTy, use the Socks5 with ProxyCap or Proxifier and start Steam using the -tcp argument by making a shortcut to Steam.exe to your Desktop (or anywhere else) and it to the start parameters.

Use VPN services. I use Astrill which is a paid service because I have other needs besides connecting to Steam, but there should be plenty of free ones out there. A good VPN can connect you to Steam with your full network capacity without any issue.

I run OpenVPN on a VM at home. A little port forwarding on the router is all it took to get set up properly. (It's free for <10 concurrent connections). I used to use DropBox to host the VM on my desktop and started it up when I knew I'd need it. Now, I have an old PC sitting there doing nothing else.

System requirements are low (512MB Ram, 1 CPU is more than you'll ever need)

Use port 443 and almost any firewall will let you connect without issues (whitelisting firewalls being the exception but they're rare outside Gov't/Defense).

Doesn't matter where I am or what I want to do, I connect to the VPN and it works.

There is a performance hit - you'll get the worst speed of your VPN server/your current location and there's a (small) latency overhead but I've been able to play CS:S in the office with a ping of < 50ms.

As an added bonus, it works for my (Android) mobile too