A lot of the time nut-based milks don't need to be refrigerated until opened. You could check to see if the kind you like is shelf-stable or comes in a shelf-stable variety.

You may be able to avoid buying a fridge entirely.

I'd go to a restaurant supply store and get a basic tri-ply steel pot. Those things take years of abuse in commercial kitchens and keep going. It'll be quite a bit more than $1 though.

Another option, if you have some patience, would be to keep an eye out at thrift stores. You'd be surprised what you find some times. I just bought a nearly new all-clad skillet for $5 at one.