There are many ways to approach this design. One way would be an Acme threaded rod at each corner with a chain drive and a single motor. Ball screws would not be the best, they are not self holding like an Acme thread is. From a cost standpoint, any type of screw lift might be the most expensive, and certainly more difficult to build. A rack & pinion would not be a good choice for this application.
The simplest and least cost way would be to lift it with a foot operated hydraulic jack, just have the jack lift a central plate that is attached to all four columns. This is a common method for lift tables. If there is a reason you need it to be powered, then an air operated hydraulic jack would be my choice, or it could be electrically operated with a small hydraulic pump.