I have been asked to correct about 1/4" of gap between the front and back of drawers and the bottoms. This chest is very old, the bottoms are 1/2" poplar stock that have been trimed to 1/4" on ends and front to fit into groves. How can this be fixed without destroying the value of the piece?
I can think of a few solutions to this problem but I'm not sure how palatable any one of them will be to you or to the owner of the chest given the age of the piece. I'm afraid I also have no idea how any of them might affect value as the furniture market is full of arbitrary standards on what is OK to do to old pieces and what is not.
The first is fairly obvious: make entirely new drawer bottoms. If you make new ones from solid wood that match the old ones in construction detail you would want to be quite certain of your dimensions and that the new stock used was very well dried and stable so that future seasonal movement doesn't pose a risk to the drawers.
A much better solution in many respects would be to skip solid wood and use 1/4" plywood, which is both stronger and more stable. Obviously you'd sidestep worries of expansion and contraction of the drawer bottom but ply may be an undesirable material given the age of the chest.
Revise the existing bottoms
In theory you could revise the current drawer bottoms by sawing them apart and glueing in a narrow piece of fresh wood down the centre. If done well the repaired bottoms would be very strong, as strong as originally in fact, but this option is the most work and I'm not sure it's worth the trouble to try to salvage as much of the original wood as possible. As well as that, without careful toning the repair will presumably be very visible.
Although you have to expand the panel by only 1/4" with the loss of wood to the saw kerf and subsequent planing to joint the edges the strip of new wood will end up being wider than this, possibly 7/16".
Bridge the gap underneath
The last option is the easiest but might be the least favoured by a traditionalist, and that would be to glue and pin in small slips of wood to whatever parts of the drawer box are needed to bridge the gap and support the existing bottom from underneath.