This is what's called a bandsaw box, and this particular one was probably created using roughly the steps below, which are slightly different from those for a more typical bandsaw box.
cut off front and back of log
cut out outer shape of drawer
cut off back of drawer
cut out compartments
glue back of drawer back on
glue front of log onto front of drawer
You don't explicitly state your question in the post, but I'm going to answer assuming that it's along the lines of
What can I do for the drawer bottoms to reduce sag?
To address your points:
I was originally thinking some straight steel rods perhaps 1/4 inch (6.4 mm) in diameter placed under the bottom oriented front to back
I'm trying to think of a ...
I wouldn't recommend trying to improvise with the wrong style of slide, because it will take longer, is likely to be more frustrating, and may not work as well in the end. That said, you do have some options:
Buy a replacement part from the original manufacturer of the bed. If the originals failed prematurely, you may be entitled to free replacement parts. ...
Yes, the 1/8" gap that you mentioned is recommended precisely to account for wood movement, and you may find that the drawers fit too tightly once they pickup moisture during the more humid summer months.
Similarly, if you had constructed the drawers during the humid summer months instead of the winter, you would want a tighter fit to begin with, because ...
It's worth noting right from the start that 3/16" plywood is very thin for the bottom of a 36"x39" drawer. You're correct in thinking that you will need to do something to prevent the bottom from sagging too much. But on the bright side, your drawer bottom doesn't need to support 50 lbs--or, more correctly, it cannot support a full 50 lbs without exceeding ...
you should be fine. the drawer sides are thin enough that expansion/contraction of them is not an issue. furthermore, although you are pinning the dimension of the drawer from the inside, the outside is free to expand - but again, it's such a small amount. solid drawer bottoms are not glued in because the bottom panel is wide enough for expansion/contraction ...
Properly speaking, you don't need any power tools to build a cabinet. (However, that would be tiresome to say the least.)
Yes to the router, no to the router table (unless you commit an entire 4x8 sheet to the table), and an additional "yes" to a circular saw with a home-made cutting guide. This picture from woodmagazine.com:
They specify 1/2" stock for ...
For dados in larger boards, a router guided by a straight edge clamped square to the board can be easier than a router table. There are a number of good jig designs, depending on how often you're going to cut the same dado, how much alignment you want the jig to do automagically, and so on.
Yes, I don't have any dado blades, anything I would use them for I use a router for. While for long rabbets the dado can be better, good set up and technique the router can do just as good.
The router is also much smaller than a table saw (a good one) and has many other uses as well.
Bottoms are planned to be made from 3mm thick hardboard.
This is the part you should probably change. If you make the bottoms from something much stouter it will self-support. You could go up to 6mm hardboard, which is significantly stiffer than 3mm, but something like 12mm particleboard/chipboard or plywood would be the superior choice. 12mm and even 18mm ...
It looks like the runner is screwed into the frame. In this case unscrewing it and adding spacers between the runner and the frame, then screwing back should provide a solution. You should probably put the spacer behind the blocks the runner is screwed to, and not between the runner and those blocks (to get more stability).