Question:
Is there a way to filter water without chemicals or a whole purification system?

Answer: you can boil the water, it takes about 10 minutes at a rolling boil to kill of all the bad stuff, i think that the water filter do a pretty good job, they are alot cheaper than a purification system and do almost the same jobYou can use a bed of granular activated carbon to filter water. But what are you trying to filter? The sink water, the shower, the whole house, or just a glass of water?

You can buy point of source filters like inline filters which use many different types of filtration media. Depending on the media will give you varying degrees of filtration. Heck you can always distill the water.