To save space, you can start seeds in soil blocks. You use a small press that you pack soil into that creates these little cubes. Once the plants sprout and start to grow you can transplant to a larger container, until it’s time to plant them in the ground.

Tomato seeds starting to sprout 
120 seeds planted in this small space 
Time to transplant
More info on soil blocks:
https://growagoodlife.com/soil-blocks-for-growing-seedlings-2/