how thick should linear rods/extrusions be

I'm considering building a printer with a crossing axis type gantry, like the ultimaker but the axis will be driven differently. The bed will be around 350 square and I'm thinking about using v slot or carbon tubes for the linear motion. They each have their perks but my main question is how thick should each be for the span I'm wanting. I think they can afford to be thinner than usual as the load is distributed over several rods or extrusions.
Before anyone says linear rails, that's a no. If I'm going to use extrusions to bolt the rails to I'd just use the extrusions and wheels. Rails are too expensive for me to build this printer using them, especially given it'll need more than the usual xy gantry.

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I think they can afford to be thinner than usual as the load is distributed over several rods or extrusions.

That depends on the load itself. Is it a moving load, then it won't be shared equally between the rods. You'd have to have each rod big enough to hold the load within the expected drop.

Fair point. The load will be a possibly direct drive extruder and light duty router (MDF). I'm thinking 10-12mm rods or 2020 extrusion. 2020 seems like plenty, I could even go smaller maybe, I'm not too sure about the rods. For the 2020 I'll be rotating the rods 45 degrees along their axis, so the longest dimensions (diagonals) line up with the direction of the load (vertical and horizontal).