Up to this day, SpaceX stores Falcon 9 rockets horizontally and there are two fundamental reasons for that. First of all, it is much easier to work on lower objects (horizontal rocket) as they test and check each booster before taking it to the launchpad. The second reason is that lower buildings are cheaper and easier to build as their gravity center is lower and the building process is easier.
However, Starship may be different. It is estimated to weight about 4400t at launch (with fuel, in storing condition it will be less, however a lot more than Falcon 9 which is about 550t). The storing system depends on the fact will they be able to develop a technology that will be able to effectively and safely transition rocket from horizontal to a vertical position. The other factor is that Starship and Superheavy are designed to have as many flights as possible. So most probably they will land and launch frequently(as we can conclude from the position of launchpad and landing pad), as a result, they will most probably be designed for being carried while vertical.