We have limited storage in our house but manage to control the toys to a degree using:
a) lidded boxes which we can stack for toys we either have a lot of or which have large pieces.
b) plastic drawer units for things which we have less of or which have very small pieces so you can get a lot in a drawer.
c) inexpensive bookshelves, for books obviously and also for big toys to sit on top of eg Playmobil Castle and Pirate ship.
d) hammock for soft toys.
Using this method we can take a box or drawer out to play with the stuff and there is an obvious container for them to learn to tidy back up into.
We try to operate a one box out one box back system so that once it is sorted we don't have to do it again for a while.
When I am tidying I sort into piles and then box/drawer/shelf them. I tend to keep a box or similar container for miscellaneous items so I can narrow down the sorting quickly. This way it is less overwhelming.
Remember you don't have to do it all at once. You can decide you are just going to pick up all the books today. Then pick up all the soft toys the next day and so on.
A cheap version of storage boxes with lids are bankers boxes. They are not transparent so you do have to label them but they do as you gather plastic boxes.
Let us know how you are getting on. I need to do my ds room after the Christmas influx and it is a bit scary. 