This has been covered to some extent before, but you have a few options. I live in an apartment so I understand where you are coming from. I play both acoustic and electric and have never had complaints from the neighbors. I actually bought a small practice amp for daytime playing on the electric, and late at night I'll either play electric unplugged, play my acoustic softly, or hook my electric up to my computer and play through a VST amp like Guitar Rig that can go to headphones or speakers on a low volume.
Your problem sounds pretty extreme though- if even playing an unplugged electric is bothering the neighbors there isn't much you can do. You must be really strumming pretty hard for them to be able to hear that through the walls. Usually the loudest part of playing an electric unplugged is the percussive slaps of the pick on the strings, so if you are playing heavily you can dial it back a bit and it shouldn't be audible to anyone outside the room.
If you want to amplify yourself a bit, the easiest option would be a guitar processor/multi-effects pedal with a headphone out, that way you could play softer and the only sound will come through the headphones.. surely that would be quiet enough to satisfy the neighbors. If you have an audio interface you can do the same thing on your computer using a VST guitar amp like Guitar Rig or Amplitube.
I am actually looking to move to a more isolated living space (a house further out of the city) so I can do more playing/recording without worrying about neighbors. If the people in your building are so intolerant, it might be worth it to consider a move if guitar is important to you, but they sound like they are just being unreasonably sensitive to the situation. Might be worth a talk with the landlord.