。
There was a dog just inside the gate.
Literal
Gate [gen-の] immediately-inside [at-に] dog [subj-が] existed.
すぐ内側 ('right inside, immediately inside') uses すぐ as a spatial/temporal intensifier 'right, immediately' + 内側 ('inside, interior side'). すぐ+position words is productive: すぐ隣 ('right next to'), すぐ前 ('right in front of').