Learn Russian: Ящик слишком тяжёлый, его нельзя поднять. - The box is too heavy, it cannot be lifted.

This is the core of how нельзя works with aspects! When you use нельзя with a perfective verb like поднять, it means the action is physically impossible to complete (the box is too heavy). If you used the imperfective поднимать, it would mean the action is forbidden (you are not allowed to lift it).

Why is the second half using его instead of он for 'it'?

The English translation says 'it cannot be lifted', using a passive voice where 'it' is the subject. The Russian sentence is active but impersonal. Literally, it means 'it is impossible to lift it'. Because ящик (box) is a masculine noun acting as the direct object of the verb поднять, it takes the accusative case form, which is его.

Could I say невозможно поднять instead of нельзя поднять?

Yes, you can. Невозможно specifically means 'impossible' and always carries that meaning, regardless of the verb aspect. However, нельзя combined with a perfective verb is a very natural, everyday way for Russians to express that something just cannot be achieved.

I've seen ящик translated as 'drawer' before. How do I know which one it is here?

Ящик is a versatile word used for a box (like a crate or packing box) as well as a drawer in a desk. You just have to rely on context. Here, a heavy box makes perfect sense, though a heavy drawer would also fit the sentence perfectly!