The password less than 10 digits could be brute-forced within days with cloud servers. Here is a demo tutorial for password breaking. The average speed is 1000 per second.
How safe is the password protecting a Monero wallet?
As the title says, how much brute force is it needed to find the password of a Monero wallet? Here's an (older) article that claims that any standard Windows password can be cracked by a cluster of 25 GPUs in just 6 hours: http://arstechnica.com/security/2012/12/25-gpu-cluster-cracks-every-standard-windows-password-in-6-hours/
I was wondering if it would be safe to send myself my Monero wallet to my addresses on public email services such as Gmail or Yahoo Mail, so that I never lose it, even if I lose my laptop and my desktop breaks.
The password hashing in Monero uses the same algorithm as mining. So you can estimate the rate that brute forcing is possible from the rate of mining. The entire mining network of Monero could brute force about 10-15 million passwords per second. Assuming strong passwords, that really isn't very many.
That still gives little to no protection against weak or especially very weak passwords however, which might give a brute force attacker a hit after only a few thousand guesses. That's easy to do on a laptop.
I dont think cloud storage is safe at all even if it is encrypted. If the password to decrypt your encrypted email is ever exposed to a keylogger, the encryption no longer protects you. Keep most XMR in cold storage, using view key to verify balances after incoming transactions if needed.
If you need to make a lot of outgoing transactions create a 2nd wallet with a smaller balance for that purpose. That way if your private key is ever exposed to a key logger less is at risk
As others have said, always use extremely strong passwords and brute force attacks to crack them will not be feasible
It's as strong as you make it. If you use a crappy password you're gonna have a crappy time when someone steals your funds. If you use a good password, you're gonna have a good time because all your moneros are safe and sound.