I have a farm shed with 1-100 W Light bulb.Recently the neutral overhead wire broke. Can I drive a ground rod and still use the light? Or do I need to run new wire?

Absolutely 100% NO, and then yes.

A ground rod serves NO role in the functionality of the electrical system. It does NOT serve as, provide, or create a "ground" as far as a circuit ground.
And it especially would not or could not serve as a current carrying neutral.

Run a new wire.

Agree 101%. If the wire broke is there a reason you can't splice a repair?

No reason other than cost .I was quoted $250 min. for a bucket truck to come out and repair.
Thanks for the replies.

$250 is a lot. If you only use the light occasionally, you can probably get creative with a solar panel, battery, and high efficiency, low voltage bulb for less money and more fun!

I'd get a direct bury cable and a shovel before I'd pay $250 for a truck.