您不能使用official image（docker pull linuxbrew/linuxbrew）有什么原因吗？它基于Ubuntu 16.04 / Xenial。
如果你必须使用Bionic（18.04），正确的安装方法是按照官方Dockerfile中的步骤进行。
但是要使您的Dockerfile工作，您需要安装ruby，创建一个非root用户，并以该用户的身份执行安装脚本。

FROM ubuntu:18.04

RUN apt-get update && \
    apt-get install build-essential curl file git ruby-full locales --no-install-recommends -y && \
    rm -rf /var/lib/apt/lists/*

RUN localedef -i en_US -f UTF-8 en_US.UTF-8

RUN useradd -m -s /bin/bash linuxbrew && \
    echo 'linuxbrew ALL=(ALL) NOPASSWD:ALL' >>/etc/sudoers

USER linuxbrew
RUN sh -c "$(curl -fsSL https://raw.githubusercontent.com/Linuxbrew/install/master/install.sh)"

USER root
ENV PATH="/home/linuxbrew/.linuxbrew/bin:${PATH}"

PS：我添加了--no-install-recommends来忽略可选的依赖项，添加了rm -rf /var/lib/apt/lists/*来删除apt-get的剩余项，从而减少映像大小。另外，添加了locales来安装UTF-8，否则brew会在运行命令时抛出警告。

新的正确方式是：

RUN apt-get update && \
    apt-get install -y -q --allow-unauthenticated \
    git \
    sudo
RUN useradd -m -s /bin/zsh linuxbrew && \
    usermod -aG sudo linuxbrew &&  \
    mkdir -p /home/linuxbrew/.linuxbrew && \
    chown -R linuxbrew: /home/linuxbrew/.linuxbrew
USER linuxbrew
RUN /bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install.sh)"

Gabriel的回答对我来说基本上是有效的，但是缺少了一个步骤。我需要将chown文件夹/home/linuxbrew/.linuxbrew给运行Homebrew的用户：

RUN apt-get update && \
    apt-get install -y -q --allow-unauthenticated \
    git \
    sudo
RUN useradd -m -s /bin/zsh linuxbrew && \
    usermod -aG sudo linuxbrew &&  \
    mkdir -p /home/linuxbrew/.linuxbrew && \
    chown -R linuxbrew: /home/linuxbrew/.linuxbrew
USER linuxbrew
RUN /bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install.sh)"

USER root
RUN chown -R $CONTAINER_USER: /home/linuxbrew/.linuxbrew