When the switch is "on" the voltage should be pretty close to zero. When the switch is off, you might read something closer to 120V. Odd small voltages probably indicate that you're reading phantom induced voltages. You meter has a very high input impedance.

Twisting the wires together and taping them is ***NOT*** a legal way to connect things. If there's not a proper fastener under all that tape, yes put a wire nut on it. MAKE SURE YOU TURN OFF THE BREAKER FIRST.

As for the two cut off wires, they probably were connected to something that has been removed. Hopefully, they don't have voltage on them because even though they're not stripped, it's still possible that the tips could touch or arc against the metal box or other conductive stuff in the box. I'd cap them off with a wire nut or something just to avoid problems.

This whole thing reeks of some previous harry homeowner doing some DIY work. You may want to have someone with a clue take a look at things just to make sure it is right. It's not possible to tell by looking at a photograph.

If that were my house, I'd hire an electrician to check out everything. With that many wonky things in one box, my money is on numerous code violations elsewhere that raise the issue of shock and fire hazards.

Thanks. I guess I'll have to do that. My house was built in 1964 and I'm amazed to find some really wonky things. I can't imagine it was all done by previous homeowners, someone had to do something right when the house was constructed, but no?. Examples include sump pump on the same circuit as garden lights, garden fountain, and all sunroom switches. Or all kitchen appliances and sockets, adjacent room switches, outdoor light, basement light all on the same circuit. There were other wonky things like phantom disconnected wires, etc.

Dennis-

in my experience houses are generally wired properly by licensed electricians when they are built. But after 50-60 years of ownership and modifications by inept homeowners one can find a lot of wonky things.

It's amazing what ignorant and careless people will do to wiring. If you look around, you'll find websites with photos and descriptions of some eyebrow-raising hack jobs that the pros have found. They're pretty entertaining.

In my own houses, some of the choice examples I've run into include twist-and-tape flying splices inside walls, parts of circuits wired with SJ rubber cord, NM nailed to baseboards through its middle, knob and tube run through holes in cold air return ducts with no protection, and NM laid on the ground to power outbuildings. You can't make this kind of stuff up.