You could close it in with drywall and then add cupboard doors of various kinds, but you'll need to make allowance for the HVAC register that is up there and willl need to be extended forward to the new front wall. Anyway, should be possible to make it all look like it was always that way.

What are you planning to store up there. You will need a ladder to get to it.

To use it for storage without having a make-shift or cluttered look, I would have closed cabinets built in.

Not sure about the height but maybe the horizontal cabinets from Ikea either with frosted glass doors or solid .