How to go about staircase drawers and drawers below stairs
Stringers are the zigzag style angled beams that hold up conventional stairs.



They are used on each side ot the staircase and sometimes in the middle as well.  They hold up the treads and the risers (the vertical boards) are attached to them.

Image 1 is basically floating treads on the stringers and using the risers as faces of drawer fronts. As the comments reflect, this is easily done when building a staircase, but quite difficult to retrofit. It also won't work if your treads are wider and there is a stringer (or the need for a stringer) in the middle.

Image 2 is a hybrid.  There are no stringers on the near face of the staircase in the area of the drawers, and none in the middle. There may be conventional stringers above that level.  It appears that there are horizontal and lateral supports built around the drawers and supporting the treads in place of stringers. Again, something that can be done when building a staircase, but not practical as a retrofit.

Image 3, based on a comment by John Smith, leaves the staircase intact, preserves the supporting stringers and gives you flexible storage possibilities.

The weight of materials in the drawers is not really a consideration unless you plan to store gold bars or some other really heavy materials. The weight of people on the stairs is much greater than any strain from linens or even books.