How can I reduce the amount of space my seeds take up without losing them? They can't be put into a solid block like ores and wheat. Any ideas?
Seeds cannot be compacted down in any way like wheat or ores are able to.
If you really need to store that many seeds, a single chest only takes two wood logs to craft; they are relatively inexpensive, so as long as you have the physical space, you can fit a very large amount of seeds into as many double chests as you need.
If space is an issue, there are also designs which allow you to pack a massive amount of storage into a small area:
With recent development, I would recommend now that they be composted into bone meal.