Anti Static Wooden Cabinet for Electronic Meters.

The meters are going to be stored in these cabinets, as opposed to being installed in these cabinets for use? (Maybe the answer doesn't really matter.)

You need to understand exactly how sensitive these meters are. At the very least, you'll want a Faraday shield around them, including the cabinet doors, with suitable conductors across the hinges and latch mechanism.

The Faraday shield can be installed on the inside surface of the cabinet if it is desired that the wood surface show on the outside for aesthetic reasons.

You'll need to decide if the Faraday shield should be solid metal foil or a metal(lic) screen depending on how sensitive the instruments are, and to what frequencies.

You may need to consider grounding the Faraday shield, and you may need more than one layer of shield (separated by an insulator) if the instruments are extremely sensitive. But, I'm heading towards overkill--I don't know what Electronic meters might be that sensitive.

There's a huge difference between electrostatic discharge, which is what I understand the OP to want to protect against, and radio frequency interference. You don't need a Faraday cage to protect soomething from static discharge.

Did I miss something?

I'm not sure if you missed something, I guess we'd have to hear from the OP. I might have gone overboard.

Still, what do you use to protect something from static discharge?

In my mind, the most important thing to do to protect from static discharge is to prevent it.

This means bringing everything to the same electrical potential before physical contact is made between the instrument and it's resting place in the cabinet. Start by maintaining relative humidity at around 50%.

Wrist/heel straps would be mandatory. Then when placing the instrument into the cabinet, I'd hook a strap to the cabinet and myself, if I'm holding it.

Some known conductive material would need to be covering the interior, or at least the shelves, and everything must grounded. (Or at the same potential if not bonded to ground)

I'd suggest steel shelves (or at least one steel shelf, maybe at the bottom) bonded to earth and bonded to a steel handle on the door. That way you get the good look and easy working properties of the wood, but anyone who opens it will get grouded before they touch the meters.
Del
(Maybe add a water jet which squirts anyone who opens the door, that should deter cats too)

Conductive paint, antistatic mats on each shelf, Meters in antistatic bags, cabinet grounding, proper training of staff to follow static damage mitigation procedures.

GA. An antistat wrist strap grounded to and inside the cabinet for retrieving meters will complete the job.