Choose an area of flooring free of any wiring or plumbing. Determine where the floor joists are and mark out the section between two joists with painter's tape. Before making cuts, cover the shoe plate of a circular saw with painter's tape to help protect the wood flooring from scratches.
Following the taped markings and cutting along the inside of the taped line, carefully begin cutting out the section of flooring for the storage compartments. Use a circular saw to start the cut for each side of the compartment, beginning with a plunge cut at the middle of the taped marking. Then use a jigsaw to finish off the corners, completing the square cutout. Take extra care not to damage the finished wood flooring while making the cuts, and remember to save the cutout sections to create the lids for the storage compartments.
Measure the openings to determine the dimensions of the box, making sure to account for space at the top for the lid to sit flush with the floor. Assemble the box inserts with plywood, wood glue and nails. Secure the box inserts at their corners using simple butt joints.
Following the manufacturer's instructions (included with the hardware), attach the hardware (the flush-mounted pulls) to the cut-out sections that will form the lids. Use wood screws to attach the boxes to the floor joists so that they sit recessed at the proper height. Carefully drop the lids into place. These can be secured to the floor with hinges, if desired. In this case, the lids were left unattached so that they could be lifted out completely, if necessary.