No table saw? Any kind of table saw? You don't need a dado blade, just a couple of passes with a full kerf blade on the table saw would do it. Three passes, four at the most, with a thin kerf.
1/4" bottom should be fine if your using Tandem glides. And if it's your own kitchen you certainly want Tandem with Blumotion or a good knockoff. I've made drawers over 30" wide with 1/2 thick sides. Tandem drawer glides are the key. As a matter of fact, the maximum thickness is 5/8" with the under mount glides.
There is another option altogether. Prefinished drawer stock. Comes typically in 8' lengths and number of standard widths up to 12", already has the slot. It's 1/2" prefinished ply that already has the top edgebanded. I've made drawers up to 34x21 with it. And these are drawers that are loaded not just with pots and pans but with dishes. I've had good success cutting half blind dovetails in it, with a sharp bit of course. To mask the ply in the the pins by applying a dark stain after assembly. The stain won't adhere well to the finish on the tails.
Yet another option would be just to source the boxes altogether. I just bought 11 completed half blind dovetail drawers in various sizes and depths up to 8". 5/8" hard maple, sanded and ready to finish. The joints look as good as I can do with my Leigh Super18 jig. Paid less than $30 per drawer. They even cut the two slots in the back for Tandem glides.
How about getting a little portable router table? You don't need much for such a non-demanding task. A Ryobi from Home Depot would do it. They have one for $99 that comes WITH a little 1.5hp router. It even has above the table height adjustment. It's a casual use tool but it would certainly make grooves for drawer bottoms.