What follows is just my experience. Your mattress and situation will be different so think of this as a sketch and use it as a jumping off point. If you have a better idea that's great. My partner only decided to post this after I was finished so it's not as complete as it might have been.
The overall plan is to cut out a middle section of both the mattress and the box springs leaving two ends for each of the same size. Then closing up the ends and sewing covers for them out of canvas and upholstery fabric. The basic idea is pretty simple but it is a bit of work and requires both a Dremel tool (or something to cut the metal springs) and a sewing machine.
I used a Sealy Posturepedic Full mattress but yours will probably differ in brand, size and construction. I'm only describing what I found and did. Be flexible as you will probably have to make adjustments depending on what you discover after you open yours. I recommend a little exploratory surgery and poking around inside before you make any major cuts. But the basic approach should work for most mattresses.
Purchasing the fabric is the most expensive part of this project. I used remnants I already had so this project cost me nothing but your situation will probably be different. Note that upholstery fabric can run $30-$40/yard so keep your eye out for sales. I also used canvas for the sides which can cut your costs greatly. You can also do one side using less expensive fabric and the other in a nice upholstery fabric and flip it over for company.
Note: Don't forget to use goggles, ear protection and gloves.
Dremel tool with a fiberglass reinforced cutting wheel
Sewing needle and upholstery thread
A lot of straight pins
hand clamps (the ones that look like industrial clothes pins)
bee's wax (to wax hand sewing thread)
Step 1: Cutting Through the Box Springs
I measured the cushions on a regular couch to find out where to cut the two ends of the box springs. I wanted an L-shaped sitting area formed by two sections but you can create any configuration you like. Look at the bottom of the box springs and see where the wood cross pieces fall. It's important to cut it so that one is left to form the new back line. This maintains the box integrity. In my case this made the bottom section slightly narrower than the mattress section on top of it. But in the end the overhang made it more comfortable to sit down and stand back up.
Cut through outer covering and padding
Use a regular pair of scissors to cut through the covering right down the center. This was my exploratory cut. See how yours is constructed before you open it up too much. Peel the covering back. You will now be able to see into the box springs. Measure the two ends and mark the cut on the springs themselves with a Sharpie pen (I used blue so I could see it better). Remember to make this mark longer so that you can bend the cut end over. You don't want the metal rod poking straight out through the fabric.
Using a Dremel tool with a Fiberglas reinforced cutting wheel cut down the line of springs (see photo). Make sure to wear eye and ear protection and gloves because the sparks smart. Hold the tool steady because any jerky movement will break the wheel. Depending on your mattress you may want to cut the springs longer (the height of the box springs) and then use pliers to fold the wire down forming the back wall of the new section. I tied the folded down wire with cotton rope to give it more support.
If it is possible cut the wood leaving a supporting crosspiece to form the back line of the new edge (see photo). If this isn't possible nail a crosspiece along the back edge.
Close up ends by pulling covering and padding back into place. Wrap it around to the bottom of the wooden crosspiece. Staple in place and trim excess. Then pull the bottom thin fabric (looks like interfacing) and staple it back on and trim excess.
Repeat on the other end of the box springs to form the two lower platforms.
Step 2: Cutting Through the Mattress
Use short pieces of thread to lessen the tangling and knot often. You don't want the thread to break and a large section of your seam to unravel. It is easier to have a second person help you hold the mattress and the seam as you sew. I also folded the seam part over to have a more finished look.
Step 3: Sewing New Coverings
Lay the fabric on the sections to measure and cut. I wanted a contrasting color for the sides so I sewed strips of brown fabric together to form one long wrap-around piece. On one short side hem a slit that will create the opening to get the cover on and off. You can see this in the third photograph on this page. I allowed the slit to wrap around the long sides about 3". You can test this to make sure the mattress will fit through this slot.
This is where all those pins come in. I anchor-pinned the brown wrap-around piece (inside out) on the mattress and laid the decorative fabric on top (also inside out). I then pinned it to the wrap-around piece. Depending on the stretch of the fabric you might want to pull it taut as you're pinning so it's not baggy when you finish. I had 1/2" seam allowance. Make sure that as you are pinning you are not pinning to the mattress itself (other than the anchor pins which you will remove later to take off the cover for sewing). I then trimmed any excess fabric. Off to the sewing machine. At the 1/2" mark I did a straight stitch all the way around. Then I went back with a zigzag around the edge. I snipped the curves to make sure that when I turned it right side out the corners were smoothly curved.
I took the cover back to the mattress (still inside out) and put it on the mattress. I flipped the mattress over and anchor-pinned the sides on to hold it in place. Then I pinned and trimmed the second side remembering to keep the fabric inside out. This is when you use the slit in the side. After you've removed the anchor pins that are holding the side panels in place slide the cover off the mattress as you would a pillow case.
Back to the sewing machine and repeat for the second side.
Slip Cover Over Mattress
Turn the finished cover right side out and slide it onto the mattress. If you made it nice and snug this may take two people. Straighten it all the way around and begin sewing up the slit.
I used a thimble and needle-nose pliers to sew the ends. I also used a glover needle that has a pyramid point which is a little stronger and a little easier to grab with the pliers. You might want to pull the seams together and clamp. I used those clamps that look like industrial clothes pins. Butt the two hemmed edges together (don't overlap) and hand stitch. Wax the thread with bee's wax to both strengthen it and help it glide through the fabric. Keep in mind that each edge is hemmed so you will be pushing the needle through multiple layers of upholstery or heavy fabric. Take lots of breaks and stretch your hands. This will take longer than you think it will. As with the mattress use shorter pieces of thread to lessen the tangling and knot often. You don't want the thread to break and a large section of your seam to unravel.
Measure and Cut Panels for the Box Springs
The idea is the same for the box springs except the bottom is just wrapped around and stapled. I didn't do a bottom panel. I also only used a heavy cotton for the top panel instead of upholstery or decorative fabric since it isn't seen. Since the box springs cover is dropped on and wrapped around there is no need for the end slit.
Pile them up, throw on some pillows and you're finished. Take a well-deserved break.