Your 120V circuits will each be on one side or the other of the 240V.
Assuming you are able to tell which is which, if you turn off all the
breakers for circuits on one side as well as any 2-pole 240V breakers, what
happens to power on the circuits on the side that remains on?

If the neutral is open then these should be dead as well. But for that to
happen you'd also have to have also a missing or bad ground at the panel or
maybe your ground is ok but the utility's grounding of the neutral on the
secondary side of the transformer could be bad.

However, with a neutral open but a ground that is present but mediocre you
might read proper voltage with nothing turned on and even have satisfactory
operation of a few lights and other small stuff but severe voltage drop
when trying to drive a heavy load.

As the others have said call the utility or a professional electrician ASAP
and be VERY careful about coming into contact with ANYTHING including what
ought to be grounded. If the neutral is open somewhere and grounding
broken or insufficient not only can the neutrals be lethally hot but it's
even possible for the grounds to be hot (less likely with conduits although
possible depending on physical construction but easily possible if the
grounds are just wires within Romex and the like).