Why not an A frame style roof
The angle will help with water runoff and aid in strength for the width of your bunker.
This is all you need.
I've noticed that a MAJOR habit on this forum is to completely ignore the question, and give suggestions of how the person posting would do it, rather than actually help with what you're doing.
Simple fact is, the weight isn't such a major concern, as long as you have it distributed properly. Mankind has been building these things since long before the advent of concrete, and if you're only going two to four feet down, this would be just fine. Just build'er solid, and keep it simple. A good 90 degree angle, using 2''x4'' with a 6'' gap between each, and a good piece of plywood over top will hold up just fine (this is probably overkill even.) as long as you have a good beam to hold the weight at the top and the bottom. It's simple geometry. Honestly, the average house in northern regions could probably hold a good foot of soil on top without any worry, as they're designed to have a few feet of snow on them... which mind you, can weigh more than dirt with some melt and freeze going on. As for support beams, if you want something bigger than a root cellar, you're going to build this a lot like a house. The only difference is that you'll need a bulkier frame, and will need a strong base to support the sides. You might also consider just making the whole thing a triangle instead of a box with a triangular roof. (This wouldn't be noticed if you built it wide enough, since the sides could be entirely partitioned off for storage space, leaving you with a rather standard blocky living space that is so popular in modern construction.) Remember, this is simply an issue of directing the weight along the structure, and accounting for compressional forces. Also, treat the entry-way just like you would when installing a window on a structural wall. As long as you shore up everything to a standard of double what you would in an above ground construction, you'll probably be fine.
The main concern with wood construction will really be moisture. Moisture will rot and soften the wood, and not just what seeps out from the earth, but what comes from anything stored inside. For this, I don't really know what to tell you, as most of my experience in underground construction has been either simple storage spaces and getaways out in the woods, made out of local material and made to last maybe a decade or two with upkeep, or old random metal containers and scrap, welded up. But as long as you give good drainage around the site (gravel, sand, and such) and waterproof the outside fine, you could probably get off with a low-power dehumidifier running inside, just like you'd put in your basement.
Now, can we see more helpful posts that actually aim to answer the questions at hand, rather than just "OMG I wants a dome! BUILD A DOME! He should totally forget whatever he's planning to do, that's actually within his own ability, and cost range, and use our idea instead." so people can actually get somewhere? Also, if you guys are really considering a SHTF scenario, why in all hell would you have something built for you, that you personally won't be able to upkeep without resources that are only available through industrialized society? What are you going to do when your concrete starts to crumble, and you don't have any available to patch it up? This guy can just go cut down a tree, mill a piece of timber, go back inside and brace the structure up... cut out the bad wood, replace it, then remove the bracing. Like it or not, if you can't be self reliant in your plans, they're doomed from the start.