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Divergence of J= - ∂ρ/∂t (equation 1)

Where ρ is the density of electric charges/ volume

J= the current density = Amperes/m2

I understand that if the divergence is not zero, the rate of change of the amount of charge is changing inside a closed control surface. But I am reading the book from Purcell, and he says:

"In a region where the current is steady div J = 0 (I understand this, simply for a steady current the same amount of charges enters and exits from the surface) together with equation 1, this tells us that the charge density is zero within that region".

I don't understand the last part.

If divJ=0, it means the same amount of charge enters and exists the surface, but there is charge inside the surface right? It does not mean the charge inside the surface is zero. Is this correct?

What I think is:

Divergence of E equal to zero ----> the net charge inside the closed surface is zero

But divergence of current J equal to zero---> the current density is constant

What I am reading is:

Divergence of J= 0 means the charge density is zero within that region.

What I don't understand is, if the same amount of charge is entering and exiting from the surface the div J will be zero but still some charge density different from zero will be inside the surface.

What do you think?