Converting my comment to an answer:
I would think the more appropriate phrasing to convey the meaning you want would be something like:
The baker never needs to buy bread.
I can't think of a common English version of this idiom off the top of my head but I think this would be more understood than how you've currently translated it. The issue with the translation in your question is that the use of the word "his" can be unclear.
the baker never buys his bread
This can be interpreted in two ways, the way you intend (I'm guessing) and another, more negative way:
The baker always has bread, so he doesn't need to buy the bread for his table.
The baker doesn't like his own bread, so he gets bread from another bakery.
My version of this statement removes the ambiguity and only allows the first interpretation.
Regardless of all of this, there's nothing in the English version that gives any implication that the baker is poor, only that his work gives him the bread he needs for the day.
I don't know the Russian saying or what the actual meaning of it is, but in English, there's a sub-definition of "bread" that implies all "food [or] sustenance", not strictly bread. It's most notably part of the Lord's Prayer
Give us this day our daily bread
So, I suppose the concept of literal bread being the bare minimum for the more figurative bread could get you a little bit of that implication of surviving on the bare necessities of life.
Additionally, if you want to be pedantic, you could argue that, technically, he does pay for the ingredients, so he does still "buy" it to some degree.
Similar concept that I thought of when thinking through this answer:
- Don't go to the barber with the best haircut.
This is more similar to the second case I mentioned above... Barbers can scarcely be expected to cut their own hair, so the one with the best haircut is clearly not the best barber... it's the person who cut his hair.