I installed a ceiling fan 3 years back. Wiring would have been difficult as it is located in a cathedral ceiling and walls didn't have any existing electrical wires.

I used a remote control for the light and fan.

Recently, it stopped working. I changed the battery and the remote control shows it works, i.e.. Light on control lights up when button is pushed.
But neither the fan or light works.

I checked the fuse box and nothing tripped.
Any ideas?

Remotes and/or receivers do good bad. You can buy a Hampton universal remote with receiver for $39 at HD or a off brand for $29. Replace both units and double check the dip switches so they match. I know you said a cathedral ceiling but you should check the electrical connection before you remove the receiver just to be sure you do have power. You're one the ladder either way.

Does the fan have pullchains?

If so, remove the remote control receiver from the fan canopy and wire the fan without it. Test the fan with the pullchains, if it works as it's supposed to then replace the remote and receiver as ballenger suggested.