Traditionally, boats were waterproofed by using Oakum which is jute rope teased into fibers, then soaked in oil and tar. It was wedged into the gaps between the planks. When it hit water, it expanded and created a waterproof seal. Great for boats, but I don't want that oil/tar smell in the house. So, here's a suggestion:
OSI QUAD sealant. They have it at Lowe's and Menard's. It's for exterior and it's mineral spirits cleanup. It dries elastic - never hardens, but I've seen recommendations for using it on flooring gaps too. You should probably use painters tape on the boards on each side so you can keep the line straight and the adjacent boards clean. That's what I intend to use. It comes in several colors including brown, beige and tan, white and clear.
Update on my floor: I've got most of it scraped (what a job) and I've decided to stain and finish the wood instead of painting. I figure I can always paint, but once I paint, I can never get back to that natural look. So it's stain/finish first, see how it holds up and if it doesn't, I can paint it later.
** Surprise! I found a 'treasure box' in the floor. A couple of planks had been cut out under where the wood stove once sat. Nothing inside but a pearl button and a bullet casing, but definitely a hiding place where I'm sure they stored their loot before banks were available. I have also run across a lot of buckshot in the cracks of the boards. Guess he filled his shells there too. Very cool! It's going fast. I'm figuring March it should be done.