How to Access styles from React?

You should use ReactDOM.findDOMNode method and work from there. Here's the code that does what you need.

var Hello = React.createClass({

componentDidMount: function() {
  var elem = ReactDOM.findDOMNode(this.refs.container);
  console.log(elem.offsetWidth, elem.offsetHeight);    
},

render: function() {
   return (
      <div ref={"container"} className={"container"}>
        Hello world
      </div>
   );
}
});

ReactDOM.render(
  <Hello name="World" />,
  document.getElementById('container')
);

jsFiddle

For React v <= 15

console.log( ReactDOM.findDOMNode(this.refs.container).style); //React v > 0.14

console.log( React.findDOMNode(this.refs.container).style);//React v <= 0.13.3

EDIT:

For getting the specific style value

console.log(window.getComputedStyle(ReactDOM.findDOMNode(this.refs.container)).getPropertyValue("border-radius"));// border-radius can be replaced with any other style attributes;

For React v>= 16

assign ref using callback style or by using createRef().

assignRef = element => {
  this.container = element;
}
getStyle = () => {

  const styles = this.container.style;
  console.log(styles);
  // for getting computed styles
  const computed = window.getComputedStyle(this.container).getPropertyValue("border-radius"));// border-radius can be replaced with any other style attributes;
  console.log(computed);
}

Here is an example of computing the CSS property value via React Refs and .getComputedStyle method:

class App extends React.Component {
    constructor(props) {
        super(props)
        this.divRef = React.createRef()
    }

    componentDidMount() {
        const styles = getComputedStyle(this.divRef.current)

        console.log(styles.color) // rgb(0, 0, 0)
        console.log(styles.width) // 976px
    }

    render() {
        return <div ref={this.divRef}>Some Text</div>
    }
}

It's worth noting that while ReactDOM.findDOMNode is usable today, it will be deprecated in the future in place of callback refs.

There is a post here by Dan Abramov which outlines reasons for not using findDOMNode while providing examples of how to replace the use of ReactDOM.findDOMNode with callback refs.

Since I've seen SO users get upset when only a link was included in an answer, so I will pass along one of the examples Dan has kindly provided:

findDOMNode(stringDOMRef)

**Before:**

class MyComponent extends Component {
  componentDidMount() {
    findDOMNode(this.refs.something).scrollIntoView();
  }

  render() {
    return (
      <div>
        <div ref='something' />
      </div>
    )
  }
}
**After:**

class MyComponent extends Component {
  componentDidMount() {
    this.something.scrollIntoView();
  }

  render() {
    return (
      <div>
        <div ref={node => this.something = node} />
      </div>
    )
  }
}