Side-storage won't send bar oil into the cylinder. The gas is the problem.

Bar oil leaking into the combustion chamber is basically not a thing. The oil tank is sealed off from the crankcase, and the only oil that moves goes through the pump to the bar. There’s no path from the oil tank to the cylinder, so leave that worry alone.

The real risk of storing a saw on its side is the fuel tank. Premix gas can seep past the carburetor’s float needle, especially if the saw sits carburetor-side down. That fuel dumps into the intake and then into the cylinder, where it can hydrolock the saw, wash the cylinder walls, and make it a nightmare to start. That’s the classic “sat all winter, now it won’t pull over” situation.

What will happen with bar oil is a mess, not engine damage. A full oil tank on its side will bleed oil out through the vent and the bar tip, so you get a puddle on the shelf. Annoying, but harmless to the saw itself.

Overnight on its side is fine. Long-term storage should be upright with the fuel drained, or at least the carburetor side up so gravity doesn’t dump gas into the cylinder.

Future You gets a saw that starts on the second pull instead of a flooded headache.