Is battery storage of mains power hard
It is certainly possible to store relatively enormous quantities of power from mains in batteries, and it's not terribly difficult (though it is expensive)! The real issue you run into is in trying to actually run anything from that power.

What you essentially do is go find a highly efficient pure sine-wave generator like these and start running your stuff off of it directly.  But if you would like to actually run say... your whole house from that battery?  You're going to run into a lot of scary things with trying to switch from mains power to your inverter (synchronization in order to do whole-house failover to backup is a whole different beast and I wouldn't recommend even trying it because you can easily hurt yourself, others, and damage equipemnt).

If you'd like to see an implementation check out Photonic Induction's setup.  Though be aware, photonic induction, though he does a lot of extremely dangerous and stupid stuff, knows exactly what he's doing and is very safety conscious.  You'll see in the video that he has an entire specialized setup in his fusebox which prevents him from connecting both mains and battery backup at the same time.

Hope that helps!