You can buy body calipers for estimating body fat percentage at home pretty cheaply. You can buy these for $17 or these for $8 on amazon.
3 points about body calipers:
1. This will still require you to have a separate body weight scale.
2. This won't measure body muscle percentage.
3. Using Body Calipers with the 3-site method on your own is pretty accurate, but it can be a bit difficult to do on yourself. Also, for getting the MOST accurate results using body calipers with the 7-site method, it will also require you to get help from someone to take measurements at several points on your body for you (though 3-site is accurate enough).
Use this Calculator for the 3-site Method
Use this Calcualtor for the 7-site Method
Video Demonstrations:
Body Calipers paired with a body weight scale is the cheaper way to measure body fat percentage and weight, but it is less convenient, especially if you are measuring your body fat percentage yourself.
2. Home Body Fat Scale
You can just use a convenient at home all-in-1 scale, that measures your body fat percentage, body weight, and body muscle percentage. I currently recommend this body fat scale because it has hand and foot sensors for better accuracy. [Note that this scale has a weight limit of 330lbs (150kg).]
For more consistent results with this method, measure yourself in the morning, after using the restroom but before eating or drinking anything.
This is the easier method to do on your own, but it becomes less accurate at very high or very low (under 10%) body fat percentages. It is also slightly more expensive than getting calipers and a regular body weight scale, though many people will still find it worth the investment.
Other Methods
Other methods are too expensive and/or inconvenient to do often as you lose fat.