I would suggest doing a rough measurement of floor space around each of the larger items of furniture since in general they can't be stacked. Smaller items of furniture such as lamps or small chairs can often be packed around or on top of larger pieces of furniture. Keep in mind depending on the type of furniture you have it might be possible to disassemble items (such as a bed frame) and then stack the pieces so that they take up a smaller space.
For smaller items, try and figure out approximately how many boxes of a certain size you will need for your belongings. Companies like Uhaul and most office supply stores often sell storage boxes that are uniform in size and can often be stacked as long as the contents are not that heavy (a good guess would be about 3-4 boxes high depending on item weight and type/quality of boxes. If you can determine the number of boxes required for your smaller items you can add them up to get a rough estimate of the area they will take up when stacked in your storage space.
Once you have a rough idea of the floor space required for your furniture and the amount of floor space required for the stacks of boxes just add the two areas for a rough estimate.
Keep in mind it is always a good idea to estimate a bit high. If your estimate of required area is very close to the size of the storage locker, it would probably be worth going with the next size up just to be sure everything fits.
One more tip: When moving furniture in a mobile storage container it is generally a good idea to separate adjacent and stacked items with a blanket or towels so that they don't rub together and damage each other during the move.