I am in the process now of reorganizing our books.
A year ago, I put some books in a box in the closet, to try that out. We have space issues too. I regretted it because they were not handy and were covered up by other heavy boxes. Even though I typed a list of titles first, I still could not remember what books I had in there. Sometimes I just knew I had a particular book and could not find it. Now I have pulled that box out and put them on the shelf. This helps me remember what I have since I am very visual. You never know when you may want to use that book that is in storage.
What kind of shelving are you using now? The best thing I did was get tall, deep, heavy shelving. Floor to ceiling shelving on a full wall would be ideal, but would not work well in our home. Money is tight so I stalked craigslist for a while and found a great set under “entertainment center” for $250 for 8′. These are heavy, high quality and thick. Dh said it was like moving a safe and he used an appliance dolly. The two 30″ bookshelves are each 16″ deep. This is great for large books. For smaller books, I put them in small baskets to fit two rows on the shelf to utilize the whole shelf space in an orderly way. The tv armoire will be repurposed to a clothing armoire by hanging a rod inside, which helped solve a closet dilemma in a bedroom. I sorted through books and got rid of some less desirable ones too.
In short, do not store books where not easily accessible. Make good use of your limited space with proper shelving. Keep only the good books that you and your family will likely use and enjoy and learn a lot from. I have no idea what your home and library look like, so this may not have been very helpful to you.