FIX: Could not connect to Steam network error

To cut a long story short, if your internet connection is not blocked in any way (such as by your ISP or your computer firewall), I've found 2 ways that work so far.

The first way worked once and then it didn't, while the other... well, let's just say that's a work in progress. I'll need to check again tomorrow to see if this 2nd Fix is more permanent than the first method.

Method 1

1. Go to Program Files OR Program Files (x86) -------- Steam ---------- delete the "userdata" folder OR if you like, you could rename or shift it to the desktop or something as backup
2. Restart Steam

Method 2 (Update: It's a permanent fix... yay!)

1. Go to Run (Start Menu --------- Run OR right click on "Start Menu" and then "Run")
2. Type in steam://flushconfig and press Enter
3. The flushing managed to boot up my Steam and I finally was able to log in.

(Source: https://support.steampowered.com/kb_article.php?ref=3134-TIAL-4638)

Hopefully, these methods help! I'll see if I can get more methods in short of reinstalling Steam, which you could try as well.

Have fun gaming! :)