So long as you and all the mosfet's leads and your meter are all at the same potential there should be no problem - so just make sure you touch everything you're using before you start and don't go walking around, mid-test. An enhancement mode mosfet should show up as a diode between source and drain (I think in a depletion mode one you'd see a low resistance there but I don't know for sure), with the gate appearing open circuit. A bipolar transistor should show up as two diodes connected back to back with the base being the anodes (for NPN) or cathodes (for PNP).
I work on an old metal tea-tray. It's not earthed, just at "me" voltage.