Analog multimeter won't calibrate to zero
I assume, since it has a zero-ohms adjustment at all, that this is an analog multimeter (one with a physical moving needle). In which case, you should try replacing the internal batteries. They are used only in resistance mode (so everything else will work), and if they are weak they will not be able to produce enough current to move the needle to full scale, which is what you are seeing.

Many meters do not have separate battery compartment doors. If you do not see one, try removing the entire back cover (if there are obvious screws with which to do so). You may also find fuses, which would need to be replaced if the current mode of the meter is not working.



The above advice is for the zero ohms point which is at the far-right end of the scale. The left-end zero in other modes — where the needle rests when no connection is made — should be adjusted using the screw or knob on the face of the meter at the pivot point of the needle. This is a mechanical adjustment of the return spring. If turning it has no effect, then the meter may be damaged, or the linkage between the adjustment screw and internal parts may have become disconnected in which case careful disassembly and reassembly may fix it.