I want to go from Washington DC to India by sea. Is it possible? How many days it will take?
Probably yes, via Europe, it should be possible, but you might have to travel parts of the journey overland. It will take a long time. You might not depart exactly from Washington DC, but I will assume elsewhere along the US east coast should be fine as well — the travel from DC to the starting point of the ocean journey will be much less than the time for the rest of the journey, anyway. Also, I'm not sure if you can disembark on a freighter in India, but perhaps you can disembark a couple of days' train journey away (still checking).
You're looking for freighter-travel. You might need to combine different freighters. They won't take a direct way, but will make many stops along the way that are not directly useful for you. For example: