A 2.5V reading could well be due to a high input impedance of the meter. My 8846 when switched to high input impedance will read anywhere between 1.5V and 6.5V. It does tend to have higher readings when the leads are plugged in.
As already stated, if the input is shorted, the reading should drop to zero.
With regards to calibration, you can check the calibration yourself if you have access to a higher spec instrument, preferably with better than 10x the accuracy of the instrument you are checking, but it is acceptable to have an instrument 3x better accuracy. You then use that instrument to verify the voltage / current / resistance being applied to the instrument being checked. You will find however that measurement technique will also start to come into play, but it is a great way to learn about the instruments and their limitations.
The high spec calibrators are really just for commercial calibration labs to get the instruments through the door and back out as efficiently as possible with recorded results. You can get just as good accuracy yourself with the right application and techniques used.
If you have to adjust calibration, then that can be a different matter. You will need the specific values at the required accuracy to follow the manufacturer's procedure and you generally find that you need all of them as you have to go through the whole calibration procedure, otherwise it will be invalid. You will also need the correct stable ambient conditions.
You can of course send it out to a cal lab, but be careful what you are getting. In my country, calibration means checking, if adjustments are required, then that is an extra charge. UKAS calibration (checks) of my 8846 costs circa £120, standard calibration checks are around £80. It has never needed adjustment, so do not know what that would cost.
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