效果演示

import 'package:flutter/material.dart';
import 'package:dio/dio.dart';
import '../config/http_headers.dart';

class HomePage extends StatefulWidget {
  HomePage({Key key}) : super(key: key);

  _HomePageState createState() => _HomePageState();
}

class _HomePageState extends State<HomePage> {
  String showText = '还没有请求数据';
  @override
  Widget build(BuildContext context) {
    return Container(
       child: Scaffold(
         appBar: AppBar(title: Text('请求远程数据')),
         body: SingleChildScrollView(
           child: Column(
             children: <Widget>[
               RaisedButton(
                 onPressed: _jike,
                 child: Text('请求数据'),
               ),
               Text(showText)
             ],
           ),
         ),
       )
    );
  }

  _jike(){
    print('开始请求数据..................');
    getHttp().then((val){
      setState(() {
        showText = val['data'].toString();
      });
    });
  }
  Future getHttp() async {
    try{
      Response response;
      Dio dio = new Dio();
      // 设置伪造的请求头
      dio.options.headers= httpHeaders;
      // 随便找的网站的地址
      response =await dio.get("https://time.geekbang.org/serv/v1/column/newAll");
      print(response);
      return response.data;
    }catch(e){
      print(e);
    }
  }
}

网站的请求头

// 打开网站F12里面复制过来
const httpHeaders = {
'Accept': 'application/json, text/plain, */*',
'Accept-Encoding': 'gzip, deflate, br',
'Accept-Language': 'zh-CN,zh;q=0.9,en;q=0.8',
'Connection': 'keep-alive',
'Content-Type': 'application/json',
'Cookie': '_ga=GA1.2.1213330956.1557299866; _gid=GA1.2.1546145805.1557299866; Hm_lvt_022f847c4e3acd44d4a2481d9187f1e6=1557299866; SERVERID=3431a294a18c59fc8f5805662e2bd51e|1557300893|1557299841; _gat=1; Hm_lpvt_022f847c4e3acd44d4a2481d9187f1e6=1557300917',
'Host': 'time.geekbang.org',
'Origin': 'https://time.geekbang.org',
'Referer': 'https://time.geekbang.org/',
'User-Agent': 'Mozilla/5.0 (Macintosh; Intel Mac OS X 10_14_4) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/73.0.3683.103 Safari/537.36'
};