On a slight tangent (no need to eject and works in develop), this may be of help to other folk looking to add their current git commit SHA into their index.html as a meta-tag by adding:

REACT_APP_GIT_SHA=`git rev-parse --short HEAD`

to the build script in the package.json and then adding (note it MUST start with REACT_APP... or it will be ignored):

<meta name="ui-version" content="%REACT_APP_GIT_SHA%">

into the index.html in the public folder.

Within react components, do it like this:

<Component>{process.env.REACT_APP_GIT_SHA}</Component>

I created another option inspired by Yifei Xu's response that utilizes es6 modules with create-react-app. This option creates a javascript file and imports it as a constant inside of the build files. While having it as a text file makes it easy to update, this option ensures it is a js file packaged into the javascript bundle. The name of this file is _git_commit.js

package.json scripts:

"git-info": "echo export default \"{\\\"logMessage\\\": \\\"$(git log -1 --oneline)\\\"}\"  > src/_git_commit.js",
"precommit": "lint-staged",
"start": "yarn git-info; react-scripts start",
"build": "yarn git-info; react-scripts build",

A sample component that consumes this commit message:

import React from 'react';

/**
 * This is the commit message of the last commit before building or running this project
 * @see ./package.json git-info for how to generate this commit
 */
import GitCommit from './_git_commit';

const VersionComponent = () => (
  <div>
    <h1>Git Log: {GitCommit.logMessage}</h1>
  </div>
);

export default VersionComponent;

Note that this will automatically put your commit message in the javascript bundle, so do ensure no secure information is ever entered into the commit message. I also add the created _git_commit.js to .gitignore so it's not checked in (and creates a crazy git commit loop).

It was impossible to be able to do this without ejecting until Create React App 2.0 (Release Notes) which brought with it automatic configuration of Babel Plugin Macros which run at compile time. To make the job simpler for everyone, I wrote one of those macros and published an NPM package that you can import to get git information into your React pages: https://www.npmjs.com/package/react-git-info

With it, you can do it like this:

import GitInfo from 'react-git-info/macro';

const gitInfo = GitInfo();

...

render() {
  return (
    <p>{gitInfo.commit.hash}</p>
  );
}

The project README has some more information. You can also see a live demo of the package working here.