Why does my 2N7000 show a gate current of more than 4mA?
Gate of a MOSFET cannot drain 5mA current. If you said 5pA I could understand, but 5mA is practically and theoretically impossible.

Either;


There is a leakage somewhere across the gate and source pins of the MOSFET.
Your multimeter has a bias.
The MOSFET is broken.
There is a wrongly done connection around the gate.


Suggestions:


Connect a gate resistor of 10\$\Omega\$ or 100\$\Omega\$. Send 1kHz, 0-5V pulses to the gate. If you can, amplify the voltage on the resistor by using an opamp difference amplifier. Observe the voltage drop on the resistor by using an oscilloscope (not with that multimeter).
If you constructed your circuit on a breadboard, reconstruct it in a different region.



  The circuit is indeed constructed on a breadboard. Which effects would show this behaviour?


Dirt inside the breadboard structure my cause leakage current.


  I’m still open for blaming the multimeter though


You should. That's probably the only culprit.


  The FET seems to do what it is supposed to. Is there a failure mode which introduces such a large leak current but makes it still work correctly?


The system will still work if there is a ghost resistance between gate and source pins which will draw 5mA current. The system will still work even if that resister were very small in value as long as the power supply can supply enough current.