Negative Feed on Electrical Lines
It's a scam, as the others say. At least consider that he, or two working together,  may have disconnected your power in order to attempt to obtain money from you.
How you deal with this is up to you, but regardless, he's talking rubbish and is 'up to no good'. If he's doing this to you then he's probably preying on any gullible and fragile people he can find. The doorman is meant to be some form of (minor) defence against problems - not a source of them. Having written the above I've convinced myself that in that situation I'd feel the need to have him dealt with by landlord and/or police




  our power went out at about 7:30 am. 


When did the doorman start work. Does he know the maintenance guy. 


  we have a negative feed problem. 


The term "negative feed' has meaning in some contexts but none in this one.


  "In an old apartment, sometimes one of the 2 live wires 'dies' 


They are alive or dead. If dead, just repair.


  and when that happens your appliances consumes power even when they are turned off. 


Rubbish / malarkey / fabrication / lies / ...
Do I make myself clear? :-)


  And that's what's called a negative feed. 


And again.


  He also said something about 1 of wires as positive and the other, negative." 


Wires on one phase can be phase and neutral / live and ground etc.
Postive and negative are DC circuit terms and suggest incompetence on the part of the scammer.  


  He then said that he can revive the dead wire by putting a device. The whole thing will probably cost us a thousand but he can do it for $250. He also said that the same situation happened to one of the tenants in the building a week ago.


Wanna buy a bridge?
Oh. Wait. Someone already offered you one in the comments :-)
ie this is a scam. And what would the landlord say about this bill. And is this man the landlord's official fix it man? and if so, for how much longer?


  His explanation was vague and theory-wise, I've never heard of such a thing. I'm a graduate of electronics and communication eng'g back in 1982 in my home country. While I know much has changed and I haven't put what I learned into practice, his explanation was highly suspicious and I immediately thought that I'm being scammed.


Am I? 

Only if you want to be.