Yes, you can. The first time I tried it, I used a plastic pot with holes drilled in it, a plastic cup for water and a piece of cloth as a growing medium. It worked pretty well. Here's what I did: I had a 10-inch pot with holes drilled in it, so I put two pots side by side, about 2 inches apart. I put the pot that was to grow spinach in the second one.
I put the pot with the plastic cup on top of the spinach pot, and put the plastic cup right over the holes in the spinach pot. Then, I put a small plastic cup with water in it next to the spinach pot. The plastic cup was a bit bigger than the pot, so I could put a few seeds in it and put the pot right over them. I put a little bit of soil in the bottom of the plastic pot, so that the roots wouldn't get dried out. Then, I put some seeds in the plastic cup, and put the pot right over the seeds. When the seeds germinate, they'll grow up through the holes in the bottom of the spinach pot and into the water. You can use any kind of seed, but I found that spinach seeds germinated the fastest.
In a couple days, the plants will be big enough to transplant into their permanent pots. I put the little seedlings in a bigger pot, and then planted them outside in my garden.