The biggest drawback and most commonly heard complaint about inversion tables has to do with storing them. They're huge and bulky, and because of this, the most convenient thing would be to simply leave them set up and in position all the time. The problem with that, however, is that an inversion table in use requires much more room than a table just sitting in a room somewhere, which means that you'd need a fairly large room dedicated just to the table if you wanted to leave it set up all the time.
Also remember that some table models don't fold up or collapse at all. Granted, these are usually your more expensive, motorized tables. Budget models tend to be both lighter weight and more portable. Many, if not most of these will fold up at least to a degree.
For those of us who have limited space and can't afford to delegate a large room just to our inversion tables, the manufacturers have invented a few solutions to make things at least a little better. The problem, of course, is that the more limited you are on space, the more challenging things become, and sometimes, partially or completely dismantling the table takes longer than the inversion itself.
The most obvious way of folding would be just collapsing the A-frame together, and in this way, you'll still get a fairly high (tall), but mostly flat contraption that can be propped against any convenient wall.
The drawback of this folding approach is that the table, when leaning against the wall, is not terribly stable. A small child playing with the backrest could easily knock it over and then get hurt. For that reason, the Body Flex Sport Inc., has introduced an interesting design change to prevent any movement of the backrest by adding a small safety lock to the design of their Body Champ inversion table. To make the folding inversion table more stable, you can just use that lock to hook the height adjusting shaft to the frame. The real pity is that this ingenious solution wasn't copied by all the major manufacturers.
You might want to slide a table so folded under one of your beds, but this can actually be a bit trickier than it sounds. First of all, the inversion table can weigh up to 80 pounds, which might be way too much for someone with a back problem to lift. The table would have the ankle locking system sticking out on one side, and the top of the backrest and usually the long handles on the other, so you'd wind up needing as much as 18 inches of clearance under the bed. You may be able to take the height adjusting shaft out (but now we are back to dismantling more and more of the table), to make it not quite as high. Side note about this: If the table you've got has a sticker (as opposed to an engraved) with height markers on it, it will probably get torn or pulled off, so be sure to mark your and your family's settings in advance!
If the above still doesn't make the table compact enough, the next step would be to take the backrest off the frame. All of the Teeter inversion tables are fitted with self-locking hooks that you'd need to push with your thumbs to release the pivot pins and the backrest. Most of the other tables have their secure hooks screwed into the frame so you'll need tools to get the backrest off in those cases. At this point, you'll have a relatively flat (if disassembled) inversion table. At this point, the only thing you'll have to worry about are the pivot arms, which can easily be bent, but if you suspect this might be a problem, you can simply remove them as well.
If you're looking for a folding inversion table that's easy to store, get a Teeter inversion table in my opinion. You can slide the height adjusting shaft nearly all the way in, and secure it in place using the spring-loaded pin and the extra hold made on the shaft for that specific purpose. You'll need to flip the backrest all the way around until it is resting against the crossbar. This way, the table will have its ankle locking system and handles sticking out from one side only, leaving the A-Frame open a little will allow for standing without leaning against anything.
Inversion tables are intended to be used on a daily basis, or if not daily, then on a very regular basis, and as is the case with oft-used equipment, the first consideration should be in trying to find a place to put it where you can leave it up all the time. If this is not possible, then begin looking seriously at storage options and design features on your table that will facilitate easier storage.