Large language models (LLMs) offer a powerful opportunity to simulate the results of social science experiments. In this work, we demonstrate that finetuning LLMs directly on individual-level responses from past experiments meaningfully improves the accuracy of such simulations. We construct SocSci210 via an automatic pipeline, a dataset comprising 2.9 million responses from 400,491 participants in 210 open-source social science experiments. Through finetuning, we achieve multiple levels of generalization. In completely unseen studies, our strongest model, Socrates-Qwen-14B, produces predictions that are 36% more aligned with distributions of human responses to diverse outcome questions under varying conditions relative to its base model (Qwen2.5-14B), outperforming GPT-4o by 15%. By finetuning on a subset of conditions in a study, generalization to new unseen conditions is particularly robust, improving by 71%. Since SocSci210 contains rich demographic information, we reduce demographic parity difference, a measure of bias, by 10.6% through finetuning. Because social sciences routinely generate rich, topic-specific datasets, our findings indicate that finetuning on such data could enable more accurate simulations for experimental hypothesis screening. We release our data, models and finetuning code.