Objects disappear when camera gets closer

I have a problem with when the camera moves closer to the objects in the scene they'll just disappear.

The grid of spheres is generated using Graphics.DrawMeshInstancedIndirect(). I had this problem before and read that this can be caused by the camera clipping plane. Since I couldn't apply a value smaller than 0.01 for the near-clipping plane I resized the Sphere grid by a factor of 1000 which is already ridiculous but it still happens. So I saw that I can change the clipping plane via script to values smaller than 0.01 which I tried with really small values which are starting to throw errors:
Screen position out of view frustum (screen pos 524.000000, 572.000000) (Camera rect 0 0 1026 577) UnityEngine.SendMouseEvents:DoSendMouseEvents (int)

Nevertheless, this was only making it slightly better for a value of (0.0000000001) which is weird. So somehow all dimensions are really messed up and the application is pretty far from redesigning everything. Does someone have any idea how to solve this? I need to zoom into the grid very close and it's not working like this. It might even be that it has nothing to do with that and I'm just trying the wrong thing.

Here is the camera inspector view, with crazy dimensions to match the bigger grid: