Hi SonningCP
Since the box has lost all recordings and you keep having to sign into all apps etc then the hard drive is probably kaput and your box will need replacing. There are usually also issues when pausing a live TV program for a minute or two, and then pressing play when the hard drive is broken.
You might be able to get it working again by doing the following:-
Try turning off the 360 by the switch on the back of the box; then on the front of the 360 box hold down both the standby/power button and the - (minus) button at the same time. Then turn on the 360 again whilst keeping hold of those buttons for a further 30 seconds.
If the standby/power button and the - button workaround doesn't solve it then try using the standby/power button and the + (plus) button at the same time instead.
You could also try the three fingered salute, which is switch off on the back of the box; then on the front hold down the plus, minus and standby/power buttons at the same time, and switch on again whilst keeping all three buttons pressed down. When it restarts you might see a message it can't acquire the channels. If so press OK and let it restart again.
If you do manage to get your box working again by trying the above then I suggest you make a note of all of your series links as you will still probably need your box replacing in the future.
Next time you do have any issues you could try performing a factory reset.
To factory reset go to Settings (the cog at the end of the Home menu) > System > Factory Reset. It's actually misnamed and only a partial system reset and not a complete factory reset. You are given 2 options
Keep Recordings
Format Disk
Choose Keep Recordings, which means you won't lose your recordings, and any planned recordings, or series links you've set up. You may have to sign into your apps such as iPlayer, Prime, Netflix and ITVX again. Also Match frame rate and Standby power consumption may go back to their default settings if you have changed them.