As far as I can remember, my family didn't havehas never had/owned our own farmland.

The people in my village own the farmland of my village togethercollectively own the village's farmland.

My village woperated as a production team.

The villagers planted wheat, sweet potatoes, cotton, corn, soybean, sesame, mung bean, sorghum, peas and so on.

You know, becauseDue to the fact that all the villagers owned the farmland togethercollectively and worked together, no one really worked hard, so the farmland didn't produce too mucha lot/enough food.

I removed 'you know' to make it sound a bit more sophisticated. Also, 'too much' signifies an abundance. In your next sentence, you say that it felt like you were starving and this implies that not enough food was produced.

That was why we often felt like we were starving.

Back then, there was no insecticide.

You can also say pesticide if you are referring more generally to pests and not just insects.

People caught pests manuallywith their hands.

I would say 'with their hands' instead of manually because manually implies that the opposite is using electronic devices. The use of pesticides/insecticides is not necessarily electric.

I remember that sometimes, I went to the cotton field with my mom to catch the pests.

When we caught the pests, we put them in a small glass bottle and reported to the leaders of the village how many pests we had caughthe number of pests that we caught to the leaders of the village.

Since then, the living conditions for the villagers have started to get betterimproved.

The villagers couldan now at least eat steamed buns made of wheat flour.

They no longer faced hunger anymore.

As far as I can remember, my family didn't have our own farmland.

Your sentence is grammatically correct, but I doubt that it says what you mean. “As far as I can remember” means “据我所知” in this context. I suspect you mean, “I am too young to recall a time when my family had its own farmland” or maybe “I’m too young to recall the time before the farmland in my village was owned and managed collectively.”

The people in my village owned the farmland of my village together.

The villagers planted wheat, sweet potatoes, cotton, corn, soybeans, sesame, mung beans, sorghum, peas and so on.

That was why we often felt like we were starving.

This means you weren’t really starving.

It was in the early eighties that the farmland in my village was assigned to every family.

This sounds like the families still didn’t own the land, but a specific plot of land was assigned to each family for it to use.

The villagers couldan at least eat steamed buns made of wheat flour.

Because you used “since then… have” in the previous sentence, it’s more natural to describe the present here (using present tense).

They no longer faced hunger anymore.