How to drill and remove inside from precious stone through small hole?
Define "tiny".

Normally a grinder is used to hollow out a hard rock. Knowing what kind of rock it is may be relevant. For example, if it is subject to fracturing, you definitely want to use a grinder and take special measures.

To hollow out a sphere manually the typical equipment would be a flexible shaft grinder with a right angle attachment. It will be a long process. Professionals can use specialized milling machines to do something like this, but such equipment costs a lot (like $100,000 or more).

You might be able to hire a machinist to do the work. Where I live they would charge maybe $300-$500 to do something like this.

Update based on OP comment: if the hole is 5mm they are almost certainly using a CNC mill to do it with a straight shank hemispheric diamond burr. Programming the mill to do an inverse spherical turn like this requires expertise. If the outside of the sphere is irregular you will need a 6-axis mill, very expensive.

The drawing below shows the basic strategy using a straight shank with a burr. Obviously a complex setup or CNC is required to control the cutter to make a perfectly spherical interior cavity.