Sure, I think you should be able to hand wash it just fine. I would use a bath tub filled with lukewarm water and mild dish detergent. Unfold it in the water, agitate it slightly with gentle hands, drain water and re-fill with plain, cold water. After rinsing, try not to crush or wring the cloth too much and air dry flat.
Even if you're very careful, some barkcloth tends to disintegrate after years of storage, but you could always save what you could and make pillows.
