Oh my God... I'm so dumb (and maybe panicked a bit because it's the first time this happened to me). First of all - my camera is quite old. And sometimes it's not so easy to open this battery cover. When camera stopped responding it was one of the first things I tried to do - open battery cover and "reset" the camera. When I pushed this little slider camera started to beep and probably cover was stuck again (as it does from time to time) and I assumed it has some kind of a block preventing users from opening it when camera is on.
When you wrote "I don't understand" I immediately realised there's no such block and opened it. Camera turned off and now is working again.
Thanks for your reply - right now I know what may happen and how to fix it so next time I probably won't panic
And I'm going to buy at least 8Gb card tomorrow so I won't have any problems with no SD card space anymore.