If you are vacuuming your carpet and you wrap the cord around your body do you become a magnet? If you wrap an active electric cord around your body, do you become an electromagnet?
 A: [Note: this answer is about a variation on this question: what if your finger is wrapped in a single conductor coil carrying current in one direction, which produces a concentrated magnetic field? This eliminates the slight red herring of the net current being zero in two parallel conductors carrying opposite and equal currents to and from the same two-terminal device, which leads to cancellation of the magnetic fields, except at very close proximity to the conductors. The question of a body part being used as the core for a proper coil is independently interesting.]
In a frame of reference in which some electric charges are moving, we observe a magnetic field. A coil of wire concentrates this magnetic field. A coil can be wrapped around some core material which can further strengthen the magnetic field depending on the degree to which it possesses a property called permeability. Materials which do this are those which become magnetized, either temporarily while the applied magnetic field exists, or permanently.
Whether or not some appendage of your body acts in this way and to what extent depends on its permeability.
Permeability can be expressed in absolute terms or as relative permeability: the ratio to the permeability of free space. If something has relative permeability of 1, then it has the same permeability as free space. (And is definitely not a magnet, permanent or otherwise).
According to Table 1 in this paper entitled "Electromagnetic Field Modeling in Human Tissue", various human tissues are all listed as having a permeability of 1: i.e. approximately that of free space.
If an energized coil of wire is wrapped around your finger, there will be a magnetic field there passing through your finger making it look like it is an electromagnet, but your finger isn't actually doing anything; the field will be the same if your finger is removed.
A: No.  The power cord on a vacuum cleaner has both supply and return conductors, which produce opposing magnetic fields.  The region of nonzero magnetic field is limited to a few cable diameters away from the cable.
Also the magnetic field due to an alternating current changes direction at 50-60 Hz, depending on your local power supply frequency.
If you wanted to turn yourself into an electromagnet you'd need to separate the hot and neutral conductors in the power cord, and wrap one around you clockwise and the other counterclockwise.  Then the fields due to the two currents would add.  You could convert to direct current if the alternation of the field direction bothers you.  However since biological tissue is not ferromagnetic, the physical effects on you are pretty tiny.
A: Okay, accuse me of having too much time on my hands, but here's what I did:

If you can't tell from the pic. I wrapped the vacuum cord around a steel bar. I turned on the vacuum and tried to pick up the screw. Absolutely nothing not even a hint of attraction so maybe BowlofRed has a point.
In case the comment gets deleted later, here is BowlofRed's comment:

The power cord has two conductors in it. The current is moving in opposite directions in each, so the net current flow through the cord is zero. No net current, no bulk magnetic field. – BowlOfRed

A: The answer is no, because the electric field of each wire cancels the field of the other wire.  But even if you wrap yourself many times with one wire and connect the ends to electric power, you will not become a "measurable" electro-magnet.  The reason for this is that the amount of ferromagnetic material in your body (blood iron, etc.), is very small.
