Yes, anyone can grow seeds at home, it's really easy.
The container size depends on how many seeds you want to sow. If you're going to sow lots as I did in the photo above, a tray, box or tin is best. If you only want to sow a few seeds, you can use a flower pot, yogurt carton, plastic cup, food container (wash it out first) or anything similar. Seed trays are usually about 1 1/2 inch / 40 mm deep, so seedlings need to be transplanted when they're tiny so that their roots have a chance to spread out and get bigger. If you use a deeper box, flower pot or container, you can allow the seedlings to get much bigger before transplanting.
You don't need to leave seeds near an open window to get oxygen! Oxygen is all around us. In fact, it makes up 19% of the air. And it'll also permeate down through the seed compost to reach the seeds. Seeds should be kept in a warm place at a temperature of at least 18 degrees C (64 F). Once they germinate, they need to be moved to a window, bright conservatory, etc. so they can grow properly. In dim light, seedlings rapidly become straggly as the stems grow long and thin as the plant searches for light.