You could buy your soap in bars like Sunlight, Fels or Ivory soap. They look like oversized bars of hand soap but are for washing clothes.
(Think of the old washing board and wash tub.)
Laundry soap in boxes is just soap that has already been powdered for you. What is in the box is mostly air.
The bars are much more compact.

Bar soap can be grated to go into a washing machine if you like, but a cheese grater does finer job than a shredder does, much more powdery.
It also helps if your soap is drier, most body soaps have stuff in them to keep them from drying out too much.
Most laundry bars don't.

The washing soda or Borax work in two ways.
They soften the water by outcompeting the calcium ions in hard water for grease and oils.
Borax, Soda, Lye and Calcium all make soap by combining with the body oils in your clothing, but the difference is that calcium soap is not water soluble and is that grey crud you know as the bathtub ring.
This means the Soda and Borax are also grabbing loose grease out of your clothes, in effect dissolving the ring around the collar.

If you have super soft water like rainwater you won't really need them because there is no calcium in it.
Check by seeing how much soap it takes to make suds.
If you need a lot of soap to get sudsy water then you have a lot of calcium (hard water) and your cloths with have that gray look from the calcium soap unless you use water softeners like Borax..

Most laundry detergents are not really soap. They are detergents with sudsing agents. They do work, but you can't really make them yourself.
Some of your shower soaps are not really soap either but are detergents instead.

A word about phosphates. You can buy stuff like Trisodium Phosphate at the hardware store. It is a superb washing chemical. They used to add it to all the laundry detergents until it was banned for that use in the 1970s.
The problem is the phosphorus in it is a fertilizer and pouring it down the drain into lakes and rivers caused algae blooms that killed the lakes and rivers by stealing all the oxygen out of them.

Also if you are using Borax or Soda don't use Chlorine bleach or vinegar in the same wash.

EDIT:
OH yes, just to add something.
I have had to wash clothes by hand often. If you ever have to do it you will appreciate a washing machine.
One of the things I learned was to put your clothes into a large tub and stomp them like you were stomping grapes for wine.
Soap is a lot easier on the feet than detergent is, and Polka music is better than Waltz music.
After getting them well stomped go over the most dirty spots like the cuffs and collar by hand. You can scrub them on a washboard or just scrub the cloth together where it is dirty.
Then back into the tub with new water and stomp them some more to rinse them.

A 5 gallon pail is to small to get your feet in, and a bathtub is to big the effectively stomp your laundry in.
Besides, bathtubs get slippery when soapy and you might slip, fall and hurt yourself real bad.
One of those Rubbermaid type totes is about the right size.

(Next edition maybe I will explain how to dry laundry)