Why doesn't Steam start?

I was having the same problem. To solve it, just delete the .steam/steam/appcache directory. You can do this from the terminal:

rm -Rf ~/.steam/steam/appcache

You have to run this every time you want to start steam until Valve fix this problem.

It stopped working 4 hours ago for my part too.

I reinstalled steam even redownloaded dota2, and it still doesn't work. I'm pretty sure that this is a steam related issue... and they will fix it pretty soon I guess.

For the time being, you can bypass this issue by removing $HOME/.steam/steam/appcache.