Can hair short an electrical circuit?
No, not at this voltage. But it could 'open' the circuit and change the impedance of the circuit in the case of a CPU socket. If hair got in between one of the pins and the contacts there could be a problem. 

Hair resistance is probably in the MΩ range, I couldn't measure anything with my meter and it goes up to 10 MΩ. 

The pins on a processor need to make contact, hair is not going to help that situation and could open the circuit. Use 'dust off' or something to get it out.