Are you considering a platform bed? Construction can be as simple as a box or boxes and an outer frame. The diagram below are for sizes which require two base units and two platforms.
The supporting box or boxes can be any substrate, and it will be the dimension of the mattress thickness and the platform thickness, usually 3/4".
For most bed heights, 18" - 20" will provide a comfortable sitting height. For integrating any drawer space in the side of the platform or base, you may figure in the possibility of one drawer of a depth enough for some storage of maybe a blanket, or comforter. For clothing, the possibility increases.
For a finished hardwood bed appearance, the supporting box and framework can be made from a hardwood plywood of your choice. For a finished mattress size of 6", and a 3/4" platform, and to stay under a 20" overall height, a 13" or less height box will work. The top corners will receive a corner gusset to maintain it's squareness and provide a point of attachment. Usually 2 cross members to the box with gusseted corners provide all the support necessary. This attachment can be as simple as using a screw or installing a "T" nut and using a bolt through the platform to the gusset. In using either screws or "T" nuts to join the two bases and mount the platform or platforms, assembly and disassembly is quick and easy for one person.
Spacing for the box under the outer frame may give room to mount lights underneath the platform and 6" to 8" is wide enough. So, the width of the box would be the dimension of the width of the outer frame, less the thickness of the frame for both sides, and the spacing desired. The length of the box would be the length of the bed, less the thickness of the frame at the foot, and the distance you want.
For example: A double (full) mattress is 54" x 75". Allowing an inch gap all around for bedding, the inner dimension of the platform would be 56" x 77". To that outer edge you would mount the outer frame, and have the top of the frame approximately 1" above the platform. This height is short enough to allow comfortable seating on the edge of the mattress. The 1" height at the head of the bed will allow for an attachment of a headboard.
Platform type beds work well with just a mattress. Of the ones I've made "polyfoam" mattresses work well, and offer good support. They can be ordered in different densities. If you order a rounded corner mattress, you can make the foot corners rounded horizontally to make getting around it safer. Or a solid corner piece with a radius will provide an "eased" corner.
As for creating a "storage" system for the underside, the bed could be made taller, the outer frame could be modified in design to install drawers. This sketch shows the basic layout of the base and platform for sizes requiring 2 bases:
Here is an example of a simple "captains" type platform bed with drawers: