I'm having the same problem as of today. "Glad" to see I'm not the only one suddenly having this problem.
Other devices on my local home network (another Mac laptop; iPhone; etc) are able to connect securely to Google and YouTube just fine.
If I reboot from my other partition (Yosemite), it also works fine.
The only problem is when I'm booted in Mavericks. (10.9.5)
It affects all browsers. FireFox has the same problem; Chrome won't load anything at all.
Tried unplugging from Ethernet and going WiFi only - no difference.
Looked at my \private\etc\hosts file - nothing unusual there, it's fine.
Turned off Little Snitch completely - no difference.
Turned off all Safari extensions - no difference (didn't expect any, since other browsers have the same problem)
Restarted my Airport Extreme - no difference.
One thing I haven't tried yet; booting from a cloned backup I made of my Mavericks partition yesterday, to see whether it'll still work.
Interested to see, as time goes on today, if (a) the problem resolves itself, (b) anyone else here has any insight into this. Very, very strange.