Welcome to the forum. As others wrote, this is a problem for the landlord, not you. However, if that weren't the case, or you just want to know what is going on, and to answer your questions:
Where were you measuring 240 volts?
If you were measuring it in the slots that are on top of each other, (as shown in your pic) that would be bad. If measuring in slots beside each other, it could possibly be a split receptacle.
Good point. However, the OP should have 120V and 120V only between:
1) Either "hot" to ground
2) Either "hot" to "neutral"
A split receptacle with 240V would only show up if measuring voltage hot to hot and two different legs were used, otherwise the voltage between the two hots would be zero.
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I moved into a newly renovated apartment on the first and the air conditioner was not working. After testing the receptacle with a multimeter it is 240V and there is no ground installed, and the AC unit requires 115V. After discovering this I tested some other receptacles and about half don't have a ground wire.
Are grounds required at every receptacle by code?
Is it dangerous for this outlet to have 240V across it?
What is the best way to go about getting this fixed?
Also, for my own curiosity would renovation of a single apartment unit require a PE to be involved? Or is this something a contractor can just come in and do on their own?
View attachment 18155
1) All 3 prong 120V receptacles are now (and for some time) require to be grounded or have GFCI protection, either thru being a GFCI receptacle, being on the load side of a GFCI receptacle, or on a circuit protected by a GFCI breaker.
2) Yes, very. That is a NEMA 5-15 receptacle made for 3 wires: one hot @ 120V (small slotted hole, can be any color of wire except green or white), one neutral (longer slotted hole, must be a whte wire) and a ground/EGC (round hole, will either be green or bare). Since all 120V appliances use that plug, anything plugged into that receptacle with an unmodified factory plug, save for something like a computer tower with a 120/240V selector switch set on 240V, will probably blow up instantly or nearly so.
3) Aside from having the landlord fix it, a licensed electrician.
Do you have any pictures with the receptacles removed?
Chances are they used to/were planning on having 240V for the AC units and now have 120V. This would require changing the receptacle from a NEMA 6-15 or 6-20 to a 5-15 or 5-20, removing one of the ungrounded conductors (white wire would be easiest/more correct) from a 2p breaker and landing it on the neutral buss (or on the neutral spot on a 1p AFCI/GFCI breaker), changing the 2p breaker out for a single pole, and ensuring the neutral has been re-identified (easiest if the white was used a hot when wired to 240V).
If the box is metal and it is AC cable or metal conduit back to the panel, a self-grounding receptacle is sufficient to complete a ground path to the appliance. I'd still prefer to pigtail to the box should the receptacle ever get loose or a replacement of the non-grounding type be used. A GFCI receptacle would be better still.
If the box is plastic or a NM type wire is feeding it, and there is only 2 wires per cable, you cannot ground the outlet w/o running new wire. In those cases, a GFCI receptacle is mandatory.