Two issues jump out:
- Your - getDatanever returns anything, so its promise (- asyncfunctions always return a promise) will be fulfilled with- undefinedif it doesn't reject
 
- The error message clearly shows you're trying to directly render the promise - getDatareturns, rather than waiting for it to settle and then rendering the fulfillment value
 
Addressing #1: getData should return the result of calling json:
async getData(){
   const res = await axios('/data');
   return await res.json();
}
Addressig #2: We'd have to see more of your code, but fundamentally, you can't do
<SomeElement>{getData()}</SomeElement>
...because that doesn't wait for the resolution. You'd need instead to use getData to set state:
this.getData().then(data => this.setState({data}))
              .catch(err => { /*...handle the error...*/});
...and use that state when rendering:
<SomeElement>{this.state.data}</SomeElement>
Update: Now that you've shown us your code, you'd need to do something like this:
class App extends React.Component{
    async getData() {
        const res = await axios('/data');
        return await res.json(); // (Or whatever)
    }
    constructor(...args) {
        super(...args);
        this.state = {data: null};
    }
    componentDidMount() {
        if (!this.state.data) {
            this.getData().then(data => this.setState({data}))
                          .catch(err => { /*...handle the error...*/});
        }
    }
    render() {
        return (
            <div>
                {this.state.data ? <em>Loading...</em> : this.state.data}
            </div>
        );
    }
}
Futher update: You've indicated a preference for using await in componentDidMount rather than then and catch. You'd do that by nesting an async IIFE function within it and ensuring that function can't throw. (componentDidMount itself can't be async, nothing will consume that promise.) E.g.:
class App extends React.Component{
    async getData() {
        const res = await axios('/data');
        return await res.json(); // (Or whatever)
    }
    constructor(...args) {
        super(...args);
        this.state = {data: null};
    }
    componentDidMount() {
        if (!this.state.data) {
            (async () => {
                try {
                    this.setState({data: await this.getData()});
                } catch (e) {
                    //...handle the error...
                }
            })();
        }
    }
    render() {
        return (
            <div>
                {this.state.data ? <em>Loading...</em> : this.state.data}
            </div>
        );
    }
}
     
    
setTimeoutor similar), using Stack Snippets (the[<>]toolbar button). Stack Snippets support React, including JSX; here's how to do one.async/awaitis part of ES2017, not ES7 (ES2016).